Topics in Dynamical systems

Oliver Knill

Preface

This is an introduction into some topics in dynamical systems. Most of the material was tought the second term 1994 at Caltech. where these notes were distributed.

This version is preliminary. There is too much material, which is covered also at other places. The literature is not yet well covered and some proofs are not self contained enough. The material needs also further condesation.

These notes are built up in several logically independent chapters. It should therefore be possible, to read any of the chapters independent of the others. The text is mathematical, which means that we state results and prove them or (in some rare cases) to indicate, where a proof can be found. We try to be as much as possible complementary to existing texts in dynamical systems and leave away things, which appear in too many places. Some of the results are indeed quite new. For example, the use of the implicit function theorem to show the existence of embedded shifts appears at many places. We want to work with careful definitions, rigorous arguments and give as much as possible full independent proofs. At some places, the need for the abstraction should become clear. On the other hand, it is now fashion to learn by examples. The aim is to prove some ”beautyful” and ”significant” theorems in the topics dynamical systems and to introduce the material which is needed to understand these. We think of a ”beautyful” theorem as a result which can also be explained also to a nonmathematician and which has not have a too technical proof. With ”significant” we mean that the theorem should either have applications or if the methods to prove it can be used in other places also. The subject itself makes it easy to fulfill the first aim. We illustate the material with many pictures as it is done in many books. However, we often give also give the source code of the programs which consists often only of a few lines. This should encourages to continue the experiments. The subject still shows many chances for open-ended exploration and research. This is the reason, why it is desirable to be close to the mathematical research fronts. A subject keeps alive with the open problems it contains. An updated list of references should make it easier to start research in one of the subjects. An other aim is to provide a global geographic overview over the subject. The one dozen landmarks given in the chapters do hopefully help as a guide in the literature. We believe that a wide picture is important in a time, where knowledge is accumulated in an immense fast rate. A deeper knowledge comes easier with independent study and research.

Pasadena, August, 1995 Oliver Knill

Contents

1 Dynamical systems in general 7 1.1 A bluffer’s guide to chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The too abstract point of view . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 The basic questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 On the definition of ”chaos” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Examples of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Hénon maps 35 2.1 Twist maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 The shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 Contractions in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 The implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Embedding a shift in the Hénon map . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Existence of the stable and unstable manifolds . . . . . . . . . . . . . . . . . 43 2.7 A strange attractor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.9 Open research problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.10 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Fractals 52 3.1 Hausdorff dimension and fractals . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Properties of the Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . 53 3.3 The Hutchinson operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 The box counting dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 The similarity dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.7 Dynamical systems on fractals . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8 Skewproduct of contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.9 The Hutchinson operator on measures . . . . . . . . . . . . . . . . . . . . . . 67 3.10 Overview over different dimensions . . . . . . . . . . . . . . . . . . . . . . . . 68 3.11 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.12 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Billiards 72 4.1 The billiard map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 The generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Existence of periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Classification of periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5 The formula of Green,Mackay,Meiss . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.7 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.8 Positive Lyapunov exponents for billiards . . . . . . . . . . . . . . . . . . . . 82 4.9 Exterior billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.10 Open research problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.11 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.12 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Rational maps 89 5.1 Embedding a one-sided shift in the quadratic map . . . . . . . . . . . . . . . 90 5.2 Special parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 The Mandelbrot set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4 The Julia sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.6 The connectivity of the Mandelbrot set . . . . . . . . . . . . . . . . . . . . . 94 5.7 Local dynamics near fixed points . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.8 The Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.9 M is connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.10 The Mandelbar set and other sets . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.11 Iteration of rational maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.12 Embedding of a one-sided shift in rational maps . . . . . . . . . . . . . . . . 106 5.13 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.14 Exercices and open research problems . . . . . . . . . . . . . . . . . . . . . . 107 5.15 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Standard maps 109 6.1 The standard map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Generalized standard maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3 The generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.5 Embedding a shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.6 The KAM theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.7 Aubry-Mather sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.8 An analytic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.9 Topological entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.10 Hyperbolic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.11 Measuring the Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . 122 6.12 Search for little elliptic ilands . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.13 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.14 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7 Cellular automata 127

7.1 Definition of cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.1.1 One dimensional cellular automata . . . . . . . . . . . . . . . . . . . . 127 7.1.2 Higher dimensional cellular automata . . . . . . . . . . . . . . . . . . 128

7.2 Topological entropy of cellular automata . . . . . . . . . . . . . . . . . . . . . 129 7.3 Examples of cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4 A map on subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.5 Properties of subshifts invariant under CA maps . . . . . . . . . . . . . . . . 136 7.6 Minimal subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.7 Cellular automata on circle subshifts . . . . . . . . . . . . . . . . . . . . . . . 140 7.8 Turing machines as dynamical systems . . . . . . . . . . . . . . . . . . . . . . 141 7.9 Dynamical systems with higher dimensional time . . . . . . . . . . . . . . . . 144 7.10 Higher dimensional time systems in other fields . . . . . . . . . . . . . . . . . 144 7.11 Open research problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.12 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8 Coupled map lattices 147 8.1 Embedding a two dimensional shift in coupled map lattices . . . . . . . . . . 148 8.2 Discrete neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.4 Open research problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.5 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.6 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9 Differential equations 154 9.1 The Lorenz system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.2 An existence theorem for differential equations . . . . . . . . . . . . . . . . . 155 9.3 Global existence of the Lorenz flow . . . . . . . . . . . . . . . . . . . . . . . . 157 9.4 The Poincaré Bendixon theorem . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.5 Equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.6 Periodic orbits and knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.7 Poincaré sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.8 The ”strange” attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.9 Reconstruction of attractors from datas . . . . . . . . . . . . . . . . . . . . . 166 9.10 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.11 Some limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.12 Other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9.13 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.14 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10 N-body problems 172 10.1 The celestial N body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.2 Vortex system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.3 The Toda system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10.4 Integration of the periodic Toda lattice . . . . . . . . . . . . . . . . . . . . . . 185 10.5 The Hénon-Heils system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.6 The integrable pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.7 Periodic orbits for periodically driven systems . . . . . . . . . . . . . . . . . . 194 10.8 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.9 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

11 Dynamical systems in number theory 202 11.1 The Collatz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.2 Pseudo random generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11.3 Encryption algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.4 The discrete Logarithm problem . . . . . . . . . . . . . . . . . . . . . . . . . 208 11.5 Recurrence and the theorem of van der Waerden . . . . . . . . . . . . . . . . 209

12 Additional exercices 211

Chapter 1

Dynamical systems in general

1.1 A bluffer’s guide to chaos

A good starting point to get the necessary knowledge for bluffing your way in life is the series of books ”bluff your way in ...” (for example [1]). We are not able to provide here a ”bluff your way in chaos” but will give so much information in order that you are able to bluff your way as a chaologist.

Since chaos science is still in fashion. From 1991-1993, 2406 journal articles and in 1993-1995 even 2451 articles usee the word chaos in the title or abstract. This means that you had to read in average 6 articles per day during that time in order to stay up to date. And then you have not yet read most of the good articles, where the word chaos does not appear.

The chaos blow reached newspapers, TV and even the film. In the nineties, the public or at least the journalists have got a bit tired of it: everybody has seen the Mandelbrot set many times on photos, posters, newspapers, TV, school and on home computers. Still it is not possible to use the insights of the ”chaologists” to predict for example the Dow-Jones index for the next week. Some newer research has therefore been devoted to the ”control of chaos”.

What is chaos?

This is not an easy question, because there are simply too many definitions for this term. One has to use finer definitions according to the categorie of dyanamical systems used. Roughly speaking, chaos means ”sensitive dependence on initial conditions”. I learned this by looking the movie ”Jurassic parc”. A famous popular story is the butterfly of Lorentz. The picture is that the motion of a butterfly can produce a perturbance of the athmosphere which results into thunderstorm within several weeks. An other story tells that the gravitational force of a little bug on Sirius can change completely the orbit of a billiard ball after it has hitten enough times the boundary of a suitable table. This statement is even proven to be ”true” if the billiard table has the shape of a stadion built by pieces of circles and straight lines. A much more simple experiment which shows the ”butterfly effect” in a wonderful way is to program the function x ↦ 4x(1 − x) in three different ways and to compare the outputs. Let’s do that in Mathematica:

T[x_]:=4 x *(1-x);   TIter[x_,n_]:=Module[{y=x},Do[y=T[y],{n}];y];
S[x_]:=4 x- 4 x^2;   SIter[x_,n_]:=Module[{y=x},Do[y=S[y],{n}];y];
U[x_]:=x+3 x- 4 x^2; UIter[x_,n_]:=Module[{y=x},Do[y=U[y],{n}];y];
{TIter[0.1,59],SIter[0.1,59],UIter[0.1,59]}
ListPlot[Table[TIter[0.1,k]-SIter[0.1,k],{k,100}]]

Note that Mathematica computes with 16 digits. This does not help! The results of the three outputs of T^{59}(x), S^{59}(x) and U^{59}(x) are completely different and illustrate that the computer does not follow the associative law during the computation. The same experiment can also be done on any programmable calculator. Because such instruments calculate with less precision, the effect can be seen even with less iterations. By the way, the map x ↦ 4x(1−x) was proposed as a random number generator by S.Ulam and J.von Neumann, who had been interested in the design of algorithms for random numbers on the first electronic computer.

A good thing to know is that if you look in the television how the balls of the numbers pool (Lotto) are falling and you shake your hand, it will affect the motion of the balls. The little gravitational change over hundreds or even thousends of miles has already an affect on the balls and will change the numbers and also the winners. Apropos television. If you have a video camara, you can film inside the picture shown by the camara. Strange patterns appear.

Many people believe that chaos has its root in nonlinearity. Let iterate such a person the cat map (x, y) ↦ (2x + y, x + y) on the torus. This is a linear map but a poor cat sitting on the torus and exposed to the dynamics will be in many little pieces after only a few iterations. The idea to put a cat onto the torus is in Arnold’s book in mechanics and the origin of the name of the map. In the next experiment, we see just one orbit of length 1000 of this map: By the way, it is a shame to use more than a line for such a program.

T[{x_,y_}]:=N[Mod[{2x+y,x+y},1]]; ListPlot[NestList[T,{0.2,0.4},1000]];

What are fractals?

A fractal is a subset of a metric space with ”fractional” Hausdorff dimension different from N. The most famous non fractal set is the Mandelbrot set: it’s Hausdorff dimension has been proven to be 2! It is not fractal according to the above definition. This result does however not affect the beauty of the pictures. A calculation of the Mandelbrot set can be done by a two line program (I tried hard to get it on one line but I did not yet succeed).

M=Compile[{x,y},Block[{z=x+I y,k=0},While[Abs[z]<2&&k<50,z=z^2+x+I y;++k];k]];
DensityPlot[50-M[x,y],{x,-2.,1.},{y,-1.5,1.5},PlotPoints->200,Mesh->False];

The name Mandelbrot was invented by Douady and Hubbard who investigated this fascinating object the first time mathematically. The first picture of the Mandelbrot set has been computed not by Mandelbrot but by Brooks and Matelski who studied Kleinian groups. Their name is however not forgotten: there is an open subset of the Mandelbrot set which is now called Brooks-Matelsky set. It is a big unsolved question, whether this set coincides with the interior of the Mandelbrot set. Douady and Hubbard have shown that this is true if the Mandelbrot set is locally connected. The later problem is the big yet unsolved question in the field. Beautyful sets are the Julia sets. They live in phase space of the dynamics opposit to the Mandelbrot set which is a subset of the parameter space. Nevertheless, the numerical computation goes very similar with a two line Mathematica program. This is the most famous Julia set, the Douady rabbit:

J=Compile[{u,v},Block[{z=u+I v,k=0},While[Abs[z]<99&&k<50,z=z^2-.12+.74I;++k];k]];
DensityPlot[50-J[u,v],{u,-1,1},{v,-1,1},PlotPoints->200,Mesh->False];

Fractals are known over 100 years now. The most famous one is the Cantor set, which is the prototype of a fractal. Did you know that it has not been found by Georg Cantor but by Henry Smith in 1875?

What is the relation of fractals with dynamical systems?

There are many relations.

1) Attractors or repellors of dynamical systems are often fractals. An example are Julia sets.

2) Subsets of parameter planes look often fractal. An example is the Mandelbrot set.

3) Dynamical systems can create self-similar fractals. An example are iterated function systems. Moreover, there is a natural chaotic map associated to each attractor of an iterated function system.

4) The closure of an orbit of a dynamical system has a fractal structure. An example is the Standard map. Here is an orbit of the standard map.

T[{x_,y_}]:=N[Mod[{2x-y+1.3*Sin[x],x},N[2Pi]]];ListPlot[NestList[T,{.2,.4},10^4]];

When speaking from fractal invariant sets of dynamical systems, it is good to know that the famous ”strange attractors” are only conjectured. Mathematically, it is in general difficult to estimate the dimension of an attractor from a specific case like the famous Hénon map, which we see next:

T[{x_,y_}]:={0.3*y+1-1.4*x^2,x};ListPlot[NestList[T,{.56,.37},10^4]]

For this choice of parameters, it is still not known (and maybe not a good question) whether we see a ”strange attractor”. A little bit gossip also here. The publication of the first papers about strange attractors had some difficulties. Henon’s paper was first not accepted because it was believed that the attractor is just a periodic orbit of large period. Also Ruelle had problems with the publication of the Ruelle-Takens paper. He had to be his own referee in order to accept the paper with Takens. Also Feigenbaum had problems to publish his results about the Feigenbaum attractor.

What is turbulence?

This is believed to be one of the big questions of our time. Even the definition of turbulence is not so clear. One could say that it is chaos for partial differentical equations. Some people believe that ”space-time chaos” is a good working definition for turbulence. Space time chaos means that not only the time evolution is chaotic but also space translation. Think of drop of blue ink falling in water. Even if time is stopped after a while, the shape of the blue color is after some time very complicated. If two space coordinates are considered as space and the third as time we have something like a chaotic dynamical system.

Is chaos useful?

The easiest reply to this question is: never ask a mathematician about the usefulness, mathematics is art! But since it is usually not the mathematician who decide how much money goes into this science, one has to reply more carefully and argue that chaos might help to understand the weather, the stock market, the health of the brain and the heart. Let us concentrate on the weather prediction: already at the beginning of this century, Poincare realised, that long time predictions of deterministic systems are problematic because systems can be chaotic and discussed already the problem of making weather predictions. The ideas were forgotten for a while and when they were rediscovered, when computers were available. An early computer study was the analysis of the Lorenz system in 1963.

Orb=NDSolve[{x'[t]==10(y[t]-x[t]),y'[t]==-x[t] z[t]+28x[t]-y[t],z'[t]==
  x[t]*y[t]-8z[t]/3,x[0]==z[0]==0,y[0]==.3},{x,y,z},{t,0,40},MaxSteps->5000];
Plo=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]} /. Orb],{t,0,40},PlotPoints->5000];
Do[Show[Plo,ViewPoint->{2k,1,1}],{k,-2,2}]

This is a simple model for the time evolution of a convecting fluid layer. Other very complex system are the stock market or neural networks like our brain. We don’t know even the exact laws for such complex systems but we can measure datas like temperature and pressure for the climate, the Down Jones index in the economic world or the electroencephalograms for the brain. It would be very interesting (and profitable) to know, if such complicated dynamical systems could be simulated by low dimensional dynamical systems. Recent research has suggested that chaos is normal in the heartbeat and contrary to expectations, it is the sickest hearts whose beat often looks most periodic. Similarly, levels of white blood cells fluctuate chaotically in healthy people but oscillate cyclically in certain patients with leukemia. There are also indications that the immune system’s method for making antibodies may involve chaotic activity. Tremors of Parkinson’s disease may arise from a loss of normal chaos in the neurological system. To answer the question about the usefulness of chaos: there is no doubt that the ideas of chaos had a tremendous stimulation for a part of science and even if direct applications are not possible, the ideas have increased the knowledge about many things.

In order to bluff your way more successfuly, we recommend to look at some of the following books [3, 6, 16, 17, 20, 24, 23, 44, 50, 51, ?, 53, 54, 55, 57] about chaos. It is no bluff however, that there are many more on the market.

Summary:

Chaos is sensitive dependence on initial conditions of dynamical systems. Fracals are sets with noninteger Hausdorff dimension. Invariant sets of chaotic dynamical systems are often fractal. On many fractals there is defined in a natural way a chaotic dynamical system. Turbulence is space-time chaotic behaviour of infinite dimensional dynamical systems like partial differential equations. Even if there are doubts that chaos theory is useful to understand complex dynamical systems in economics, medicine, atmospheric sciences, it stimulated the whole scientific work of many people and leaded to progress which would probably not have been obtained without this change of view.

1.2 The too abstract point of view

We do here a very general remark about dynamical systems, which will neither be used later nor do we think they are very useful. The remark should indicate that the concept of a dynamical system can be taken very general so that it can be found at many places in mathematics. We look at the subject from the widest possible point of view. In these notes we will however do the extreme opposite and look at concrete examples instead.

A dynamical system is a pair (G, X), where X is an element in a category of objects and G is a subgroup of the automorphism group Aut(X) or a semi-subgroup of the endomorphism semi-group End(X).

Examples.X is a set and G is a subgroup of all permutations on X. X is a topological space and G is subgroup of all homeomorphisms of X. X is a group and G is a subgroup of all automorphisms of X. X is a probability space and G is a subgroup of all automorphisms of X.

The set of possible pairs (G, X) forms again a category. (One could derefore again define a dynamical systems on this new categorie. One could also go further and iterate this.)

It turns out that the notion of a dynamical system, the construction of a in general much more complicated category from a given one is also useful to understand the original categorie. This point of view can be seen as a reformulation of Klein’s Erlanger program.

A main problem in mathematics is to classify the objects of a given mathematical structure up to isomorphisms. Examples are the categories of manifolds, knots, algebraic curves, rings, finite simple groups, topological dynamical systems, polynomial differential equations. One would like to find enough invariants, which allow to distinguish the nonisomorphic objects from each other. Having an invariant, this invariant defines equivalence classes of objects. A complete set of invariants is the moduli space of the object. It labels the isomorphy classes of the structure. An important problem is to understand the map from the categorie of objects into the moduli space, which is given by the equivalence relation. For example, one would like to know, how a one-parameter curve of structures is moving in the moduli space. If there is a region in the space of objects which is mapped into the same point of the moduli space, one speaks of structural stability. The ultimate goal is the construction of the moduli space of a structure and to find a translation between this space and the space of objects. Helpful are here invariants like numbers, sets, groups or other objects, which depend only on the isomorphy class. The classification problem is untractable for many categories and there is in general no hope to find the moduli space. By narrowing the categorie, which is done by restricting the set of objects and imposing more structure, the moduli space can sometimes be computed. Already difficult is in general to understand one specific object. Interesting are often the most simple nontrivial objects or examples which appear as models in physics or other sciences. Looking at specific examples can be interpreted as shooting into the moduli space of dynamical systems without even knowing how it looks like.

We begin with an approach to dynamical systems which is complementary to what will follow later. In the next chapters, we explore part of the subject by examples. We think however, that the importance of the dynamical concept within mathematics is best seen when looking from very far away onto the structures. Also this section is independent of the rest and it is no problem to skip it.

A semigroup is a set with a composition ∘ : G × G → G satifying associativity a ∘ (b ∘ c) = (a ∘ b) ∘ c. Examples are groups, N^{d}, (R^{+})^{d} the set of finite ASCII words with the composition word1 ∘ word2 = word1word2, the set of files on a UNIX computer with composition file = file1 ∘ file2, where file is obtained by the UNIX command cat file1 file2 > file.

Definition. A semigroup is acting on a set X if there exists for each g ∈ G a map T_{g} : X → X satisfying T_{g∘h} = T_{g}T_{h}. If the semigroup has a zero element 0, one requires that T_{0}(x) = x for all x ∈ X.

Definition. A general dynamical system is a semigroup G acting on a set X.

Remark. We will seldom need such a general structure but take G = N or G = R^{+} or G = Z or G = R. Sometimes, however, it is useful to look also at more general semigroup actions like G = N^{d}, G = (R^{+})^{d} with d > 1. If G = N, the semigroup action is determined by a map T = T_{1} = T : X ↦ X. The system (X, T) is then called a discrete dynamical system. If G = R^{+}, we have a one-parameter family of maps T_{t} on X satisfying T_{s}T_{t}(x) = T_{s+t}(x) and the system (T_{t}, X) is called dynamical system with continuous time.

The above definition of a dynamical system is in some sense too general. Until now, we are on the level of set theory and no other of the interesting mathematical structures is involved. In order to get interesting questions, some structure has to be imposed on the space X and to require that the transformations T_{g} respect this structure.

A rough classification of mathematical structures is the division into algebraic, topological, measure-theoretical, geometrical and complex-analytic structures. Examples of algebraic structures are groups, rings, fields, modules, vectorspaces. A topological structure is given by a topology (an algebra of subsets satisfying some axioms) which is often given by a metric. A measure theoretical structure is given by a σ-algebra of subsets and a countably additive function m on this algebra which is called a measure. A geometrical structure is imposed if X is a manifold. One has a complex analytic structure if X is a complex analytic manifold. Having one or several of these structures on X, we have to require that the dynamics respects this structure(s) because else, this structure is irrelevant for the dynamical system. In the algebraic case, we want T_{g} to be a homomorphism, in the topological case, we want T_{g} to be continuous, in the measure theoretical case, we want T_{g} to be measurable, in the geometric case, we want T_{g} to be smooth, in the complex-analytic case, we want T_{g} to be analytic or meromorphic.

For completeness, we give the definitions of the mathematical structures, we were talking about.

• An algebraic structure on a set X is a composition, which is a map ∘ : X × X → X. The composition can satisfy some axioms like for example associativity. Often, there are two compositions (·, +) on X, and the compatibility between these structures is again defined by some axioms. Examples of sets with algebraic structures are groups, rings, fields.

• A topological structure is given by a set U of subsets of X which satisfy the axioms ∅ ∈ U, U_{1}, U_{2} ∈ U ⇒ U_{1} ∩ U_{2} ∈ U. V ⊂ U ⇒ ⋃_{V ∈V} V ∈ U. The set U is called a topology on X. The elements in U are the open sets. A map f : X → X is continuous, if f^{−1}(U) ∈ U, it is a homeomorphism if it is bijective and the inverse f^{−1} is continuous. Often, a topological space is defined by a metric d.

• A measure theoretical structure on X is given by (i) family of subsets U of X which satisfy the axioms ∅, X ∈ U, if V is a countable or finite subset of U, then ⋃_{V ∈V} V ∈ U, ⋂_{V ∈V} V ∈ U and (ii) a map m : U → [0, 1] satisfying m(X) = 1 m(⋃_{V ∈V} V) = ∑_{V ∈V} m(V), where V is a countable family of pairwise disjoint sets in U. The sets in U are called measurable sets, the function m is called a probability measure on X. A map f : X → Y between two probability spaces (X, U), (Y, V) is called measurable, if f^{−1}(V) ⊂ U, it is called an isomorphism, if f is invertible and f, f^{−1} are both measurable. This isomorphism turns out to be too restrictive and one replaces it by isomorphy mod 0: there exist subsets X′ ⊂ X and Y′ ⊂ Y of measure one, such that f : (X′, U ∩ X′, m) → (Y′, V ∩ Y′, n) is an isomorphism.

• A geometric structure on a metric space X is given by a countable family U of open subsets U covering X and bijections ϕ_{U} : U → F_{U}, where F_{U} are open subsets in R^{n}. All the maps ϕ_{UV} = ϕ_{U} ∘ϕ_{V}^{−1} defined on ϕ_{V} (U ∩V) should have an open image and they are required to be arbitrary many times differentiable (this is called smooth). The sets U ∈ U are called charts. The set U is an atlas. The maps ϕ_{UV} are called overlap maps. A metric space endowed with a geometric structure is called a smooth n-manifold. A map f : (X, U) → (Y, V) is called smooth, if for all U ∈ U, there is V ∈ V such that f(U) ⊂ V and such that f_{UV} = ϕ_{V} ∘f ∘ϕ_{U}^{−1} is smooth. A map f : (X, U) → (Y, V) is called a diffeomorphism, if f is bijective and if both f, f^{−1} are smooth.

• A complex analytic structure on X is a smooth 2n-manifold such that if we identify C^{n} with R^{2n}, the overlap maps are analytic. The manifold (X, U) is then called a analytic manifold. An analytic map f between two analytic manifolds is a diffeomorphism such that the maps f_{UV} = ϕ_{V} ∘ f ∘ ϕ_{U}^{−1} are analytic.

Remarks:

1) The above classification is not canonical. Some dynamical systems have different structures involved. Look at the example of a continuous group homomorphism T on the compact topological group T^{d}. Here, all the structures are involved: X is a group, a compact metric space, an analytic manifold and a measure space with respect to Haar measure.

2) The definition of a dynamical system is not yet the most general one. One could define a dynamical system as a two-valued relation (correspondence) acting on a one-valued relation (set).

3) Our rough classification of mathematical structures is motivated by functional-analytic considerations. To a set X one can associate the algebra of all bounded functions on X, to a measure space X one can associate the algebra of all measurable bounded functions, to a topological space, one can associate the algebra of all continuous maps on X, to a manifold X one can associate the algebra of all C^{∞} functions, to a Riemann surface X one can associate the algebra of all analytic or meromorphic functions on X. In all these cases, the algebras are commutative and the semigroup action on the algebra. This leads to the generalisation of the notion of dynamical systems to noncommutative spaces by considering an endomorphism or automorphism of a general Banach algebra. Such noncommutative mathematics is believed to be relevant when working on a quantum mechanical level. Many people believe that noncommutative mathematics is one of the most important topics in the next century.

Mathematical structures

For all the constructions, one starts with a set X and imposes some additional structure like an operation on X or a set of subsets of X. The structure satisfies some axioms. All the structures have the notion of homomorphism or isomorphism.

Algebraic structure:

\underline{Semigroup} (X, ∘):

composition: map ∘ : X × X → X

associat.: ∀x, y, z ∈ X, (x∘y)∘z = x∘(y∘z).

\underline{Group} (X, ∘): semigroup: (X, ∘) zero: ∃0 ∈ X, ∀x ∈ X, 0 ∘ x = x ∘ 0 = 0 inverse: ∀x ∈ X, ∃y ∈ X, x ∘ y = 0

homomorphism: f : (X, ∘) → (Y, +), ∀x, y ∈ X, f(x ∘ y) = f(x) + f(y) ismorphism: f bijective, f^{−1} homomorphism

Topological structure

\underline{Topological space} (X, U): set U of subsets of X ∅, X ∈ U ∀U_{1}, U_{2} ∈ U, U_{1} ∩ U_{2} ∈ U ∀V ⊂ U, ⋃_{V ∈V} V ∈ U

\underline{Metric space} (X, d):
metric: map d : X × X → [0, ∞)
∀x, y ∈ X, d(x, y) = 0 ⇔ x = y.
∀x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z).
∀x, y ∈ Z d(x, y) = d(y, x).

continuous map: f : (X, U) → (Y, V), f^{−1}(V) ⊂ U. homeomorphism: f bijective, f^{−1} continuous. isometry for metric spaces: f homeomorphism, ∀x, y ∈ X, d(f(x), f(y)) = d(x, y).

Measure theoretical structure:

\underline{Probability space} (X, U, m): set U of subsets of X, ∅, X ∈ U ∀{V_{i}}_{i∈N} ⊂ U, ⋃_{i∈N} V_{i} ∈ U, V_{i} \ V_{j} ∈ U measure: map m : U → [0, 1] ∀{V_{i}}_{i∈N} ⊂ U, ∀_{i,j}V_{i} ∩ V_{j} = ∅ ⇒ m(⋃_{i} V_{i}) = ∑_{i} m(V_{i}) m(X) = 1.

measure preserving map: f : (X, U, m) → (Y, V, n), f^{−1}(V) ⊂ U, ∀V ∈ V, m(f^{−1}(V)) = n(V) isomorphism: f bijective, f^{−1} measure preserving. isomorphism mod 0: ∃X′ ⊂ X, ∃Y′ ⊂ Y, m(X′) = n(Y′) = 1 f : X′ → Y′ is an isomorphism.

Geometrical structure:

\underline{d− Manifold} (X, U, ϕ): metric space (X, d) atlas: U countable set of open sets ⋃_{U∈U} U = X homeomorphism: ϕ_{U} : U → F_{U}, F_{U} ⊂ R^{d} open, overlap maps: ϕ_{UV} = ϕ_{U} ∘ ϕ_{V}^{−1} : F_{UV} := ϕ_{V} (U ∩ V) → F_{V U}. ϕ_{UV} : F_{UV} → F_{V U} is smooth

smooth map: f : (X, U) → (Y, V) ∀U ∈ U ∃V ∈ V, f(U) ⊂ V, f_{UV} = ϕ_{V} ∘ f ∘ ϕ_{U}^{−1} is smooth. diffeomorphism: f bijective, f^{−1} smooth.

Analytic structure:

\underline{Analytic d-manifold} (X, U, ϕ): (X, U, ϕ) 2d-manifold, identify R^{2d} with C^{d}. ϕ_{UV} : F_{UV} ⊂ C^{d} → C^{d} is analytic

analytic map: f : (X, U) → (Y, V), f is smooth map and f_{UV} is analytic. analytic diffeomorphism: f diffeomorphism, f^{−1} analytic.

1.3 The basic questions

Like for any category of mathematical objects, there is the notion of isomorphy for dynamical systems. We have seen below, how homomorphisms and isomorphisms was defined for algebraic, topological, measure theoretical, geometrical or analytic structures. A dynamical system (Y, H) is homomorphic to (X, G), if there exists a surjective homomorphism ϕ : X → Y, a surjective homomorphism ψ : G → H such that for all x ∈ X, g ∈ G

ϕ ∘ T_{g}(x) = T_{ψ(g)} ∘ ϕ(x) .

They are called isomorphic, if ϕ : X → Y and ψ : G → H are both isomorphisms. For memorizing the condition of homomorphy or isomorphy, the following diagram can be useful.

X  ^{ϕ  \overline{}}Y

[figure: T_{g}; T_{ψ(g)}]
X  \overline{} ; ϕ Y

Let I be any mathematical object like for example a topological space or an algebraic object. An invariant for a family X of dynamical systems is a map X → I which is constant on each subset of X consiting of isomorphic systems.

A subsystem of a dynamical system (X, G) is a dynamical system is a dynamical system (Y, G) of the same type obtained by restricting the semigroup action on a subset Y ⊂ X satisfying T_{g}(Y) = Y for all g ∈ G.

Important examples of subsystems in the case G = N or G = Z are finite G-invariant sets called periodic orbits.

What are the problems when studiing dynamical systems? The questions are very similar for any mathematical structure like for example for groups, natural numbers, manifolds, topological spaces. For the mathematical structure of dynamical systems, these problems are believed to be very difficult and even untractable. It is however important to keep these basic problems in mind.

• Decision for isomorphy One of the most important problems is to decide, if two given dynamical systems (G, X), (H, Y) are isomorphic. For doing so, it is often useful to construct invariants which do not change when taking isomorphic dynamical system. If two systems have different invariants, they can not be isomorphics.

• Decomposition into smallest subsystems An other important problem is to decompose a system (G, X) into (its smallest) subsystems (G, X_{i}) and to understand, classify or count these subsystems.

• Decision if a given system is a subsystem of an other one. Related to the above problems is the problem to decide, if a given dynamical system (Y, H) can be found as a subsystem of an other system (X, G) in the sense that there exists a subset Z ⊂ X which is invariant under the dynamics G and such that (Z, G) is isomorphic to (Y, H).

• Classification of all nonisomorphic systems. An other general problem is to classify all the nonisomorphic dynamical systems for a class of dynamical systems. The congruence classes of nonisomorphic systems is called the moduli space for the class of dynamical systems.

Summary

A dynamical system is a semigroup G acting on a space X where X has some additional algebraic, topological, measure theoretic, geometric or analytic structure. The case G = N, Z is called a discretetime systems and the case G = R^{+}, R is called continuous time systems.

Like for algebraic, topological, measure theoretic, geometric or analytic structures, there is the notion of homomorphy or isomorphy for dynamical systems. Basic questions are to find invariants in order to distinguish dynamical systems up to isomorphy, to decompose a system into its subsystems, to decide, if a given system can be found as a subsystem of an other system or to find a moduli space which labels all nonisomorphic dynamical systems.

1.4 On the definition of ”chaos”

The attribute ”chaotic” is unhappy and should be used very carefully in a mathematical context. The reason is that there is not a canonical definition for this word. The general informal meaning of ”chaos” is ”sensitive dependence on initial conditions”. This implies practical unpredictability of the flow. Evenso the system is completely deterministic, it is practically impossible predict the evolution for all times. I remind you of the map T : x ↦ 4x(1 − x). Any program running with 16 digits accurancy gives already for T^{60} unreliable results. Evenso, if we would compute with an unbelievable accuracy of 10^{1}0 digits (storing one of such a number would already use the memory of a whole workstation), we could not tell, what is T^{10000000000}(0.1).

In any category of dynamical systems, there is in general a whole scale of different definitions and it is better to use the finer terminology. We are going to give a list of notions which are sometimes used for the term ”chaotic”. The list is not complete and it is not at all important to know it. It should however tell you that the word ”chaos” has too many mathematical definitions.

A) Topological dynamical systems

A topological dynamical system (X, T, d) is a homeomorphism T of a compact metric space (X, d). Known attributes for chaotic motion are:

• Transitiv. For all open nonempty sets U, V, there exists n ∈ N such that T^{n}(U) ∩ V ≠ ∅.

• Minimal. For every closed, T invariant subset Y of X, we have either Y = ∅ or Y = X.

• Uniquely ergodic. There exists only one T − invariant Borel probability measure on X.

• Strictly ergodic. Uniquely ergodic and minimality.

• Distal. The closure of {T × T (x, y) | (x, y) ∈ X^{2}} is disjoint or equal to {(x, x) | x ∈ X} ⊂ X × X.

• Sensitive dependence on initial conditions. There exists δ > 0 such that for all x ∈ X, ϵ > 0, there exists y ∈ X, d(x, y) < ϵ and n ≥ 1 with d(T^{n}x, T^{n}y) ≥ δ.

• Expansivity. There exists ϵ > 0 such that for all x ∈ X the ϵ- stable set W_{ϵ}^{+}(x) = {y | d(T^{n}x, T^{n}y) ≤ ϵ∀n ≥ 0} of x satisfies W_{ϵ}^{+}(x) = {x}.

• Topologically mixing. U, V open, nonempty, ∃m ∈ N, ∀n > m ⇒ T^{n}(U) ∩ V ≠ ∅.

• Hyperbolicity. There exists ϵ > 0, δ > 0, 0 < λ < 1, C > 0 such that for all x ∈ X

d(T^{±n}x, T^{±n}y) ≤ Cλ^{n}d(x, y), ∀y ∈ W_{ϵ}^{±}(x)

and for all x, y ∈ X, d(x, y) ≤ δ,

|W_{ϵ}^{+}(x) ∩ W_{ϵ}^{−}(y)| = 1 .

• Chaotic in the sense of Devaney ”transitive” and ”a dense set of periodic orbits”. (”sensitive dependence on initial conditions”, follows from this if the system is not itself periodic.)

• Chaotic in the sense of Li-Yorke There exists an uncountable, T -invariant set K and δ > 0 such that for all x, y ∈ K, lim inf_{n→∞} d(T^{n}x, T^{n}y) = 0 and lim sup_{n→∞} d(T^{n}x, T^{n}y) ≥ δ.

• Chaotic in the sense of Lasota. The set K on which T is chaotic in the sense of Li-Yorke is generic in X.

A quantity measuring chaos for topological dynamical systems is the ”topological entropy”.

B) Measure theoretical dynamical systems.

An abstract dynamical system (X, T, A, m) is an isomorphism T of a probability space (X, A, m).

• Ergodic. m(T^{−n}(Y) ∩ Z) − m(Y)m(Z) → 0 in the Cśaro sense.

• Totally ergodic. For all n ≥ 1, T^{n} is ergodic.

• Weakly mixing. |m(T^{−n}(Y) ∩ Z) − m(Y)m(Z)| → 0 in the Césaro sense.

• Mixing. m(T^{−n}(Y) ∩ Z) − m(Y)m(Z) → 0.

• Lebesgue. There exists a closed U_{T} invariant subspace E of L^{2}(X), such that ∑_{n≥0} U_{T}^{−n}(E) = L^{2}(X) and ∩_{n≥0}U_{T}^{n}(E) = {1}.

• Kolmogorov. There exists a T invariant subalgebra B ⊂ A with ⋁_{n≥0} T^{n}(B) = A, and ∩_{n≥0}T^{−n}(B) = A.

• Bernoulli. There exists a measure space (Y, ν) such that (X, T, m) is isomorphic to the shift σ on (Y^{Z}, ν^{Z}).

A quantity measuring chaos is the ”metric entropy”.

C) Smooth dynamical systems.

A smooth dynamical system is a diffeomorphism of a smooth compact d-dimensional manifold.

One can use the attributes for topological dynamical systems. For any invariant measure one can use the attributes for metric dynamical systems. Especially interesting are measures which are absolutely continuous with respect to the volume measure. The discipline in the intersection of ergodic theory and smooth dynamical systems is sometimes called differentiable ergodic theory or smooth ergodic theory. Attributes using the richer structure are.

• Expandivity. There exists n ≥ 1 such that for all x ∈ M and u ∈ R^{d} |DT^{n}(x)u| > |u|.

• Nonuniform hyperbolicity. All the Lyapunov exponents λ_{1}(x) ≤ λ_{2}(x) ≤ λ_{d}(x) are different almost everywhere.

• Hyperbolic. There exists a compact invariant set K and an uniform hyperbolic splitting of the tangentspace T K.

• Anosov. The whole compact manifold is hyperbolic.

• Axiom A. The set of non-wandering points K is hyperbolic and has a dense set of periodic orbits.

Quantities measuring chaos are ”topological entropy” or ”metric entropy”.

Summary:

Dynamical systems appear everywhere in mathematics, nature and applications. For example: the formulation of a Turing machine as a dynamical system shows that every calculation done in mathematics is an orbit of a dynamical system.

There is no canonical definition of chaos. According to the categorie of dynamicals systems, there exist many versions and degrees of chaos. It is therefore better, not to use the word chaos too loosely in a mathematical context.

1.5 Examples of dynamical systems

We give now a random list of dynamical systems. Also without too much explanation, the examples should illustrate, how rich the structure of a dynamical system is. The explanation is probably in most cases too short in order to understand what is going on. It is no problem, if the reader is puzzled and lost in that catalogue. Most of the examples are not yet understood anyway. We will meet some of the examples in more detail.

A) Discrete time dynamical systems:

• Permutations: Let T be a permutation of a finite set X. A subalgebra of the algebra of subsets defines a topology on X. T is continuous, if it leaves the algebra invariant. A measure is a map m : X → R^{+} satisfying ∑_{x} m(x) = 1. Finite permutations can approximate already quite well real systems. Following an idea of Rannou, we can look at permutations modelling maps on the torus.

M=800; P=N[Pi];g=1.7*M/(2P);
T[{n_,m_}]:=Mod[{n+m+Floor[g*Sin[2P n/M]],m+Floor[g*Sin[2 P n/M]]},M];
ListPlot[NestList[T,{41,43},5000]];
ListPlot[NestList[T,{401,80},1000]];
ListPlot[NestList[T,{401,150},1000]];

Approximations by finite dynamical systems exist also in other contexts and there are different topologies on different classes of dynamical systems, in which there exists a dense set of periodic transformations.

• Group translations: Let X be a group. For every α ∈ X let T be a group translation x ↦ x + α. This is a Z action. If X is a topological group, T is a homeomorphism. A group translation leaves invariant a natural measure on X, the Haar measure. A specific example T : T^{2} ↦ T^{2}, (x, y) ↦ (x, y) + (α, β) mod1.

T[{x_,y_}]:=Mod[{x+N[E],y+N[Pi]},1]; ListPlot[NestList[T,{.3,.4},2000]];

Given a real number α. Consider the dynamical system on the torus T = Rmod1 given by x ↦ αx(mod1). One knows that for almost all a, the orbit of the point 1 is uniformly distributed on the torus. Examples like the golden ratio show that one does not always have uniform distribution, because in that case, the orbit converges to 0. There is no specific example of an irrational α known, where the sequence of points α^{n}xmod1 are uniformly distributed. Specifically, one does not know for example, whether 3^{n}/2^{n} is uniformly distributed modulo 1. Such questions have connections with Fourier series (see the book of Salem).

T[x_]:=Mod[3x/2,1]; ListPlot[NestList[T,1,10000]];

• Bernoulli shifts. Let I be a finite set with the discrete topology and X be the compact topological space I^{Z} with the product topology. The homeomorphism T : x_{i} ↦ x_{i+1} is called the two sided Bernoulli shift. If we take X = I^{N} instead, we have the one sided Bernoulli shift. If I is a measure space, X is in a natural way a measure space and T is measurable leaving invariant the product measure. If I = {1, 2, …, 6} with equal distribution, there exists a subset of X of measure 1 such that x_{i} gives random numbers produced by a fair dice. More generally, one defines subshifts of finite type also called topological Markov chain by taking a |I| × |I| matrix A with entries in {0, 1} and to take the subset X_{A} of X given by all sequences satisfying [A]_{x_{n},x_{n+1}} = 1.

The one-sided Bernoulli shift on the alphabet I = {0, 1, 2, …, 9} can be computed on the computer by x ↦ 10x(mod1) on the interval [0, 1).

T[x_]:=Mod[10*x,1]; P[x_]:=Floor[10*x];P[NestList[T,N[Pi-3,100],150]]

But this is not a good way to simulate this dynamical system. A better way is to take a random number generator.

• Group automorphisms. Let X be a group and T be a group automorphism. An example is the group T^{d} where a group automorphism is given by a linear map represented by an invertible matrix M(d, Z). A famous example is Arnold’s cat map on T^{2}. Here, we show a three dimensional version of this map which is a map on T^{3}.

R:=Floor[5*Random[]]; A:={{R,R,R},{R,R,R},{R,R,R}}; B=A;Det[B]
T[{x_,y_,z_}]:=Mod[{{0,2,3},{1,3,3},{1,2,1}}.{x,y,z},1];
S=NestList[T,{.223,.334,.4},100];
Show[Graphics3D[Table[Point[S[[i]]],{i,Length[S]}]]];

• Interval maps. A well known example of a smooth map is the logistic map x ↦ a·x(1−x) on the unit interval [0, 1]. Here a is a parameter in [0, 4]. This family of logistic maps has been studied by Feigenbaum.

Famous is the bifurcation diagram, which shows how the attractor of the map depends on the parameter c.

T[{c_,x_}]:=N[{c,c x(1-x)}];
Ob[c_]:=Module[{p1=Last[NestList[T,{c,0.5},40]]},NestList[T,p1,100]];
ListPlot[Module[{s={}},Do[s=Join[s,Ob[3.+k/100]],{k,100}];s],Axes->False]

One often illustrates the dynamics for a fixed c as follows:

c=3.7; T[x_]:=N[c*x(1-x)]; L[x_]:=Line[{{x,x},{x,T[x]},{T[x],T[x]}}]
S=NestList[T,0.4,30]; G=Plot[{T[x],x},{x,0,1},Axes->False,Frame->True];
Show[{G,Graphics[Table[L[S[[i]]],{i,Length[S]}]]}];

• Circle maps. The most famous example of a circle map is the circle standard map x ↦ x + α + β sin(x), where α and β are parameters.

Here is the graph of the map for different parameters β.

f[x_,c_]:=Mod[N[x+N[Sqrt[2]]+c*Sin[2Pi x]],1];
Plot[f[x,0.1],{x,0,1}];Plot[f[x,1.0],{x,0,1}];

We can do the same as in the logistic map and compute the attractor in dependence of the parameter.

T[{c_,x_}]:={c,Mod[N[x+N[Sqrt[2]]+c*Sin[2Pi x]],1]};
Ob[c_]:=Module[{p1=Last[NestList[T,{c,0.5},40]]},NestList[T,p1,200]];
ListPlot[Module[{s={}},Do[s=Join[s,Ob[0+k/200]],{k,200}];s],Axes->False];

• Monotone twist maps. Examples are Hénon type maps on the plane R^{2} given by (x, y) ↦ (V (x) − y, bx), where 0 < b < 1 and V is a potential. Standard type maps on the cylinder or the torus given by (x, y) ↦ (x + y + V (x), y + V (x)). Billiard maps are maps on the annulus defined by playing billiard in a smooth convex table.

Here is an illustration of a billiard map at the table x^{4} + y^{4} = 1.

<<"Graphics`ImplicitPlot`"
Size[{a_,b_}]:=a^2 + b^2; Normed[ww_]:=ww/Sqrt[Size[ww]]
Curve=x^4+y^4-1; Inside[{a_,b_}]:=(Curve /. {x->a, y->b}) < 0;
Grad=Normed[{D[Curve,y],-D[Curve,x]}];
T[{P_,v_}]:=Module[{H=P,L=P,m}, While[Inside[H],H+=0.01 v];
  While[Size[H-L]>10^(-4),M=(H+L)/2;If[Inside[M],L=M, H=M]];
  w=Grad /. {x->L[[1]],y->L[[2]]}; {L,Normed[2 (v.w) w-v]}];
Orb=NestList[T,{{0,.2},{.707,.707}},30];
Show[Graphics[Line[Table[Orb[[i,1]],{i,Length[Orb]}]]],
     ImplicitPlot[Curve==0,{x,-1,1}],PlotRange->{{-1,1},{-1,1}}];

• Rational maps. A rational map is a map on the Riemann sphere given by z ↦ P (z)/Q(z), where P, Q are polynomials with no common zero. Special cases are polynomial maps z ↦ P (z). The quadratic family z ↦ z^{2} + c. Recently the study of meromorphic maps has begun. Further generalisations and active regions of research are correspondences or analytic maps in higher dimensions. One can also look at polynomial maps in any ring. Take for example the ring R[\sqrt{7}] or the ring of quaternions and iterate z ↦ z^{2} + c in this field. There are a lot of unexplored systems.

• Cellular automata: Given a finite set I. A continuous map T on I^{Z^{d}} commuting with the shifts is called a cellular automaton. Such a map is defined by a finite rule I^{n} → I. The most famous system is the game of life which is a two-dimensional cellular automaton.

Here is an example of a one-dimensional cellular automaton:

d=7; Init[n_]:=Table[Floor[d*Random[]],{n}];
T[a_]:=Mod[RotateLeft[a]+a+RotateRight[a]+a*RotateLeft[a],d];
Picture[n_,m_]:=NestList[T,Init[n],m];
ListDensityPlot[Picture[100,100],Mesh->False,ColorOutput->RGBColor];

• Random walk: The random work on a lattice Z^{d} can be viewed as a deterministic system on the space {1, …, 2d}^{Z} × Z^{d}. The map is T : (x, y) ↦ T_{x_{0}}(y), where x_{0} ∈ {1, …, 2d} gives one of the 2d directions, in which the walk goes. One is interested in properties which hold for almost all points in the space {1, …, 2d}^{Z} which labels the paths.

Here we see an example of a random walk in two dimensions:

T[{x_,y_}]:={x,y}+{Cos[N[2Pi]*Random[]],Sin[N[2Pi]*Random[]]};
Show[Graphics[Line[NestList[T,{0.,0.},1000]]],
  Frame->True, FrameTicks->None, AspectRatio->1]

• Collatz map: This is a measure preserving map on the group of dyadic integers. It extends the map n ↦ n/2 if n is even, n ↦ 3n + 1 if n is odd for which it is not known if every number is attracted to the periodic orbit 1 ↦ 4 ↦ 2 ↦ 1. This example shows that dynamical systems have relation with number theory.

The next program computes a graph which shows, how many times, one has to iterate in order to reach the attractor. The conjecture is that the graph is bounded.

T[n_]:=If[OddQ[n],3n+1,n/2];
L[n_]:=Module[{i=1,m=n}, While[m != 1, m=T[m];i++];i];
ListPlot[Table[L[n],{n,1000}]]

• Pedal map: This is a map on the space of triangles. One is interested only in the shape of the triangles. If x is a triangle, T (x) denotes the pedal triangle which has as vertices the feet of the altitudes of x. This example shows that dynamical systems have relation with planimetry.

• Conway map: Given a finite set Y of points in the plane. Draw all the connections of two points and take as the new point set T (Y) the union of Y with the set of intersections of lines. A problem of Conway is: does every orbit have an accumulation points with respect to the Hausdorff metric?

• Hutchinson map: Take finitely many contractions f_{i} on the plane and look define in the space of compact sets the map F : K ↦ ⋃_{i} f_{i}(K). This map is used to generate fractals as the fixed oint of F.

Here is Barnsley’s fern:

f={Function[x,{{ 0.000, 0.000}, { 0.000,0.270}}.x+{ .5  ,      0}],
   Function[x,{{-0.139, 0.263}, { 0.246,0.224}}.x+{ 0.57, -0.036}],
   Function[x,{{ 0.170,-0.215}, { 0.222,0.176}}.x+{ 0.408, 0.089}],
   Function[x,{{ 0.781, 0.034}, {-0.032,0.739}}.x+{ 0.1075,0.270}]};
U:=Module[{u=Random[]},If[u<0.02,1,If[u<0.17,2,If[u<0.3,3,4]]]];
g[x_]:=f[[U]][x]; start:=Last[NestList[g,{0,0},100]];
ListPlot[NestList[g,start,10000], Axes->False];

• Turing machines. These are maps on Y = X × Z = {0, 1}^{Z} × A, where A is a finite set of states and X = {0, 1}^{Z} is the tape. According to the value of (x_{0}, a) the value of x_{0} is replaced, shifted and the state a replaced by an other state. Turing machines are interesting dynamical systems because anything which can be computed can also be computed by a Turing machine. In general, any program running on a computer is a dynamical system.

Here is the busy beaver for a turing machine with 5 states

F[{state_,symbol_,position_}]:=Module[{newsym,move,newstate},
If[state==1 && symbol==b, newsym=m ; move=-1; newstate=2];
If[state==1 && symbol==m, newsym=m ; move=-1; newstate=1];
If[state==2 && symbol==b, newsym=m ; move= 1; newstate=3];
If[state==2 && symbol==m, newsym=m ; move= 1; newstate=2];
If[state==3 && symbol==b, newsym=m ; move=-1; newstate=1];
If[state==3 && symbol==m, newsym=m ; move= 1; newstate=4];
If[state==4 && symbol==b, newsym=m ; move=-1; newstate=1];
If[state==4 && symbol==m, newsym=m ; move= 1; newstate=5];
If[state==5 && symbol==b, newsym=m ; move= 1; newstate=0];
If[state==5 && symbol==m, newsym=b ; move= 1; newstate=3];
{newstate,newsym,position+move}];
T[{state_,band_,position_}]:=Module[{newstate,newsym,newpos,newband=band},
{newstate,newsym,newpos}=F[{state,band[[position]],position}];
newband[[position]]=newsym;
If[newpos>Length[band],newband=Append[newband,b]];
If[newpos<1,newband=Prepend[newband,b];newpos++];
{newstate,newband,newpos}];
NestList[T,{1,{b},1},10]

• Renormalisation maps Dynamical systems on some space of dynamical systems is called renormalisation. The famous Feigenbaum renormalisation map is defined as follows: Let M be the space of continuously differentiable even maps on the interval [−1, 1] such that f(0) = 1, ∀x ≠ 0, xf′(x) < 0. The renormalisation map T is defined on as subset D(T ) of M and has the form

T (f)(x) = −\frac{1}{a}f^{2}(−ax), a = −f(1)

The subset D(T ) is chosen so that T (f) ∈ M. The fact that this map has a nontrivial fixed point explains many observations for individual dynamical systems (see [33]).

B) Continuous time dynamical systems.

• Lorenz system: This is an ordinary differential equation in R^{3}.

ẋ = −10x + 10y, ẏ = 28x − y − xz, ż = −8/3z + xy .

A similar system is the Roessler system

ẋ = −y − z, ẏ = x + 0.2y, ż = 0.2 + xz − 5.7z .

Here we see the Roessler attractor:

RungeKuttaStep[f_,y_,y0_,dt_]:=Module[{k1,k2,k3,k4},
  k1=dt N[f /. Thread[y->y0]     ];k2=dt N[f /. Thread[y->y0+k1/2]];
  k3=dt N[f /. Thread[y->y0+k2/2]];k4=dt N[f /. Thread[y->y0+k3  ]];
  y0+(k1+2*k2+2*k3+k4)/6];
Orbit[f_,y_,y0_,t_,d_]:=Module[{q=y0,s={}},
  Do[q=N[RungeKuttaStep[f,y,q,d]];s=Append[s,q],{Round[t/d]}];s]
Orbit[{-y-z,x +.2 y,.2+x z-7.8*z},{x,y,z},{5.0,0.,0.4},40.,0.03];
Show[Graphics3D[{Line[%]}],Axes->Automatic]

• Periodically driven mathematical pendulum. The system

ẍ + cẋ + sin(x) = γ cos(ωt)

is the periodically driven physical pendulum. The differential equation ẍ+cẋ+x = γ cos(ωt) is called the van der Pool oscillator.

• The Ueda system is given by

ẍ + kẋ + x^{3} = γcos(t) .

For special parameter values k = 0.1, γ = 12, the system has a strange attractor called Ueda attractor or Japanese attractor. A very similar system is the Duffing equation

ẍ + kẋ − x + x^{3} = γ cos(t) .

Orb=NDSolve[{x''[t]+.1*x'[t]+x[t]^3-12.*Cos[t]==0,
 x[0]==0.,x'[0]==1.3},x,{t,0,400},MaxSteps->50000];
ParametricPlot[Evaluate[{x[t],x'[t]} /. Orb],{t,0,400}];

• Henon-Heils system. A famous example of a Hamiltonian systems q̇ = H_{p}, ṗ = −H_{q} is the Henon-Heils system

H(p, q) = \frac{1}{2}(p_{1}^{2} + p_{2}^{2} + q_{1}^{2} + q_{2}^{2}) + q_{1}^{2}q_{2} − \frac{1}{3}q_{2}^{2} .

The Störmer problem

H(p, q) = \frac{1}{2}(p_{1}^{2} + p_{2}^{2}) + \frac{1}{2}(\frac{1}{q_{1}} − \frac{q_{1}}{(q_{1}^{2} + q_{2}^{2})^{3/2}} )^{2}

for energies 0 < E < 1/32 is an example of physical interest because it describes the motion of electrons and protons in the van Allen radiation belts. There are other Hamiltonian systems like H(p, q) = \frac{1}{2}(p_{1}^{2} + p_{2}^{2} + q_{1}^{2} + q_{2}^{2}) + q_{1}^{2}q_{2} − q_{1}q_{2}^{2} Contopoulos,Barbanis system H(p, q) = \frac{1}{2}(p_{1}^{2} + p_{2}^{2}) + q_{1}^{4} + q_{2}^{4} + 2γq_{1}^{2}q_{2}^{2} Caranicolas-Vozikas system H(p, q) = \frac{1}{2}(p_{1}^{2} + p_{2}^{2}) + \frac{1}{2}(q_{1}^{2}q_{2}^{2} + γ(q_{1}^{2} + q_{2}^{2})) Sahos-Bountis system which are appealing because of the simplicity of their Hamiltonians.

• Toda system. This is a finite or infinite-dimensional system given by

ȧ_{n} = a_{n}(b_{n+1} − b_{n}), ḃ_{n} = 2a_{n}^{2} − 2a_{n−1}^{2}, n ∈ Z .

If there exists N ∈ N with a_{n+N} = a_{n}, b_{n+N} = b_{n}, one can show that the system is conjugated to a linear flow on a finite dimensional torus. One says, the system is integrable.

• ABC flow. The ABC flow is a flow on the manifold T^{3} defined by

(ẋ, ẏ, ż) = (A sin(z) + C cos(y), B sin(x) + A cos(z), C sin(y) + B cos(x)) ,

where A, B, C are nonzero constants.

RungeKuttaStep[f_,y_,y0_,dt_]:=Module[{k1,k2,k3,k4},
  k1=dt N[f /. Thread[y->y0]     ];k2=dt N[f /. Thread[y->y0+k1/2]];
  k3=dt N[f /. Thread[y->y0+k2/2]];k4=dt N[f /. Thread[y->y0+k3  ]];
  N[Mod[y0+(k1+2*k2+2*k3+k4)/6,2Pi]]];
Orbit[f_,y_,y0_,t_,d_]:=Module[{q=y0,s={}},
  Do[q=N[RungeKuttaStep[f,y,q,d]];s=Append[s,q],{Round[t/d]}];s];
a=13;b=4;c=7;
R=Orbit[{a*Sin[z]+c*Cos[y],b Sin[x]+a Cos[z], c Sin[y]+b Cos[x]},
  {x,y,z},{.6,.4,.5},40.,0.03];
Show[Graphics3D[Table[Point[R[[n]]],{n,Length[R]}],Axes->Automatic]];

• Hopf flow: This is an integrable flow on S^{3} is given by the differential equation in C^{2}

ż = iz, ẇ = iw

Consider it as a differential equation in R^{4}. It leaves the sphere |z|^{2} + |w|^{2} = 1 invariant. This is an example of a differential equation on a nonflat manifold.

Because of graphic problems in R^{4} we don’t give the flow.

• Orszag-McLaughlin flow: The Orszag-McLaughlin flow is given by the differential equation

ẋ_{i} = a · x_{i+1}x_{i+2} + b · x_{i−1}x_{i−2} + c · x_{i+1}x_{i−1} ,

where the index i is taken modulo n with n ≥ 3. If one takes a + b + c = 0 and abc ≠ 0, this system preserves the Lebesgue measure and leaves invariant the spheres

S_{r} = {∑_{i=1}^{n} x_{i}^{2} = r} .

• Heat flow: Let Δ be the Laplacian on a manifold M. The PdE u̇ = Δu gives an evolution on some suitable function space.

• Schrödinger flow. Given a Schrödinger operator H = −Δ + V. Define ihu̇ = Hu. Examples are the quantum mechanical harmonic oscillator H = −\frac{h^{2}}{2m}D^{2} + mω^{2} \frac{x^{2}}{2} which has the simple eigenvlues E_{n} = hω(n + 1/2) belonging to eigenfunctions ψ_{n} such that ψ_{n}(t) = e^{ihtE_{n}}ψ_{n}(0). An other explicitly solvable system is the Hydrogen atom V (r) = Ze^{2}/r having eigenvlues −2^{2}Z^{2}/(2an^{2}) belonging to eigenfunctions ψ_{nlm} of multiplicity n^{2}.

• Navier-Stokes equation. This is a PdE differential equation for two functions u (velocity) and p (pressure) on a bounded domain Ω in R^{d}. u(t), p(t) is determined by the differential equation

u̇ + (u∇)u = νΔu − ∇p, div(u) = 0

where u(x, t) = β(x, t) on the boundary of Ω. ν is the viscosity.

• Vortex flows. A model for the Euler equations discribing a fluid in the plane is given by a Hamiltonian system discribing finitely many vortices.

RKStep[f_,y_,y0_,dt_]:=Module[{k1,k2,k3,k4},
  k1=dt N[f /. Thread[y->y0]     ];k2=dt N[f /. Thread[y->y0+k1/2]];
  k3=dt N[f /. Thread[y->y0+k2/2]];k4=dt N[f /. Thread[y->y0+k3  ]];

  y0+(k1+2*k2+2*k3+k4)/6];
LastPoint[f_,y_,y0_,{t1_, dt_}]:=Module[{z=y0},
  Do[z=RKStep[f,y,z,N[dt]],{i,1,Round[N[t1/dt]]}];z]
Variab[n_]:=Table[z[i],{i,n}];
Diffeq[n_]:=Table[N[1/(2 Pi I)]*
  Sum[(z[i]-z[Mod[j-1,n]+1])/(Abs[z[i]-z[Mod[j-1,n]+1]]^2+0.0001),
  {j,i+1,i+n-1}],{i,n}];
Flow[c0_,t1_,dt_]:=Module[{l=Length[c0]},
  c=Variab[l];h=Diffeq[l];LastPoint[h,c,c0,{t1,dt}]];
Init[n_]:=Table[Random[Complex],{n}];
Pict[z_]:=ListPlot[Table[{Re[z[[k]]],Im[z[[k]]]},{k,Length[z]}],
  PlotRange->{{-3,3},{-3,3}},Axes->False];
Movie[NumbPart_,NumbPict_,TimeInt_]:=Module[{c=Init[NumbPart]},
  Do[Pict[c];c=Flow[c,TimeInt,0.01],{NumbPict}]];
Movie[4,16,0.1]
Movie[100,16,0.1]

• Retarded differential difference equations: These are differential equations of the form

ẋ = F (t, x(t), x(t − r)) .

They are actualy ordinary differential equatons on a Banach space. Examples are ẋ = −αx(T − 1)(1 + x(t)) on the line ẋ = c sin(x(t − 1)) on the circle ẍ + bẋ = c sin(x(t − 1)) on the cylinder or

ẋ = −x(t − 1)y(t) − z(t), ẏ = x(t − 1)x(t) − z(t), ż = x(t) + y(t)

on the sphere S^{2}.

C) Discrete time systems ↔ continuous time systems.

• Time one map. Continuous time systems can be reduced to discrete time systems by a taking the time 1-map. This corresponds to a stroboscopic picture of the flow.

• Poincaré map: Sometimes, it is possible to put a define a hypersurface in the space which the flow crosses transversallically, giving a so called Poincare map. This give often more information than the stroboscopic map and has the advantage that the dimension is reduced by one.

• Suspension Sometimes it is useful to suspend discrete time systems in order to get a continuous system having the discrete time system as a Poincare map.

Hénon maps

2.1 Twist maps

Given b ≠ 0 and a continuous map V′ : R → R being the derivative of a differentiable map V called the potential. A map on the plane R^{2} defined by

T : ( \begin{matrix}x \\ y \end{matrix} ) ↦ ( \begin{matrix}V′(x) − by \\ x \end{matrix} ) ,

is called a twist map on the plane.

Examples: (the examples show that one can get immortal by studdiing such maps because each map is related with a mathematician or physicst working on such a map).

V′(x) = x^{2} + c, V′(x) = 1 − a · x^{2}, Hénon map
V′(x) = 1 + a · |x|  Lozi map
V′(x) = Sign(x − a) · \sqrt{|x − a|}  Martin map
V′(x) = \sqrt{a^{2} + x^{2}}  Colin de Verdière map
V′(x) = 2x + γ · sin(x)  Chiricov map
V′(x) = 2x + 4 · arg(1 + k · e^{−ix})  Suris map
V′(x) = γ · |x − \frac{π}{2} (mod 2π) − π| − \frac{π}{2} Wojtkowsky map

The Jacobian of a differentiable map T : R^{2} → R^{2} T (x, y) = (f(x, y), g(x, y)) is defined as

DT (x, y) = ( \begin{matrix}f_{x} & f_{y} \\ g_{x} & g_{y} \end{matrix} ) .

A differentiable map T : R^{2} → R^{2} is called conservative if | det DT (x, y)| = 1 for all (x, y) ∈ R^{2}. It is called dissipative if | det DT (x, y)| < 1 for all (x, y) ∈ R^{2}. If det DT (x, y) > 0 for all (x, y) ∈ R^{2}, then the map is called orientation preserving.

A twist map T with twice differentiable V has the Jacobian

DT (x, y) = ( \begin{matrix}V′′(x) & −b \\ 1 & 0 \end{matrix} ) .

It is dissipative if |b| < 1, conservative, if |b| = 1 and orientation preserving, if b > 0. The above examples of twist maps are often studiied in the conservative case.

A map T : R^{2} → R^{2} is called invertible, if it is bijective which is equivalent to the existence of a map S : R^{2} → R^{2} such that S ∘ T (x, y) = T ∘ S(x, y) = (x, y) for all (x, y) ∈ R^{2}.

A twist map is invertible if b ≠ 0. Its inverse is then

T^{−1} : ( \begin{matrix}x \\ y \end{matrix} ) ↦ ( \begin{matrix}y \\ b^{−1} · (V′(y) − x) \end{matrix} ) .

We define recursively T^{0} = Id and T^{n}(x) = T ∘T^{n−1}(x) for n < 0. For an invertible map, we define analoguously T^{−n}(x). The orbit of an invertible map T is the sequence (T^{n}(x))_{n∈Z}. A point (x, y) belongs to a periodic orbit of T if there exists n ≥ 1 with T^{n}(x, y) = (x, y) The smallest number n satisfying this identity is called the period of the orbit.

A set Y ⊂ R^{2} is called T -invariant, if T (Y) = Y. Every orbit of a point (x, y) is invariant. If T is continuous, also the closure of every orbit is invariant. In order to understand the dynamics of a twist map, we want to decompose R^{2} into invariant sets and understand the dynamics on these invariant sets. Because we work in a topological (and not in a set theoretical) set-up, we want to find the closed or even compact invariant subsets of the twist maps. The restriction of the twist map to a compact invariant subset is a topological dynamical system, which is by definition a continuous map or a homeomorphism of a topological space. The simplest compact invariant sets are the periodic orbits.

Two topological dynamical systems (X_{i}, T_{i}) are called conjugated or isomorphic, if there exists a homeomorphism S : X_{1} → X_{2} satisfying T_{2} ∘ S(x) = S ∘ T_{1}(x). A topological dynamical system (X_{1}, T_{1}) is called embedded in a second topological dynamical system (X_{2}, T_{2}), if there exists a closed T_{2} invariant subset Y of X_{2} such that (Y, T_{2}) is isomorphic to (X_{1}, T_{1}).

2.2 The shift

Given a finite set I with cardinality |I| > 1. The set I is called alphabet. Call X = I^{Z} the set of all biinfinite sequences

X = {x = (…, x_{−2}, x_{−1}, x_{0}, x_{1}, x_{2} …) | x_{i} ∈ I} .

This set X becomes a metric space with the metric d(x, y) = 2^{−N}, where N is the largest non-negative integer such that x_{i} = y_{i} for all |i| < N. A metric space is compact, if every sequence x_{n} ∈ X has an accumulation point in X. The map T (x)_{n} = x_{n+1} is called the shift on X.

The shift is a prototype of a ”chaotic” system. It satisfies most of the attributes of chaos existing for topological dynamical systems. We introduce two such notions. A topological dynamical system is said to have sensitive dependence on initial conditions, if there exists δ > 0 such that for all x ∈ X, ϵ > 0, there exists y ∈ X with d(x, y) < ϵ and n ≥ 1 with d(T^{n}x, T^{n}y) ≥ δ. A topological dynamical system is called transitive, if for any two open sets U, V ⊂ X, there exists n ∈ N, such that T^{n}(U) ∩ V is not empty.

A topological dynamical system is called chaotic in the sense of Devaney, if

(i) it has senseitive dependence on initial conditions (ii) it is transitive. (iii) periodic orbits are dense. It is actually known that (i) follows from (ii) and (iii) [2].

We will check in the exercices that the shift has these properties.

There is defined a natural probability measure m on X = I^{Z}. It is defined as follows. Take as the measurable sets the smallest σ-algebra U, which contains the open sets. Define the measure m by defining m(Y) = |I|^{−k} for cylinder sets

Y = {x ∈ X | x_{n_{1}} = i_{1}, x_{n_{2}} = i_{2}, …, x_{n_{k}} = i_{k}} .

It is a basic fact in measure theory that this function m can be extended uniquely to a measure on U. The shift T leaves that measure invariant. It is an isomorphism of the probability space (X, U, m). The measure theoretical dynamical system (X, T, m) is called a Bernoulli shift.

A measurable function f : (X, U, m) ↦ R is also called a random variable. Two random variables f_{1}, f_{2} are called independent, if

m{f_{1}(x) ∈ A} · m{f_{2}(x) ∈ B} = m{x | f_{1}(x) ∈ A, f_{2}(x) ∈ B}

for all measureable subsets A, B ⊂ R. Exercice: Define F (x) = x_{0} and F_{n}(x) = F (T^{n}x). Show that F_{n} are independent random variables. We see that a shift produces random sequences for almost all x ∈ X. The probability that F (T^{n}x) = i_{1} is |I|^{−1} independent of the values of F (T^{k}x) for k < n.

Examples: For I = {head, tail}, the random variables F_{n} describe the outcome of successive throwing coins, for I = {1, 2, 3, 4, 5, 6}. In other words, the dynamical system is playing dice.

2.3 Contractions in metric spaces

A sequence x_{n} in a metric space (X, d) is called a Cauchy sequence, if for all ϵ > 0, there exists n > 0 such that for all k, l > n, d(x_{k}, x_{l}) ≤ ϵ. A sequence x_{n} converges in X if there exists y ∈ X such that x_{n} → x for n → ∞. A metric space (X, d) is complete if every Cauchy sequence in X converges.

Examples of complete metric spaces:

(1) X = R^{d}, d(x, y) = \sqrt{∑_{i=1}^{d}(x_{i} − y_{i})^{2}}. (2) X = R^{d}, d(x, y) = max_{i=1}^{d} |x_{i} − y_{i}|.

(3) If (X, d) is a complete metric space, then any closed subset is a complete metric space.

(4) Any compact metric space is complete because the set of accumulation points of the Cauchy sequence is then by compactness not empty and can by the Cauchy property consist only of one point.

A map ϕ : X → X is called a contraction, if there exists λ < 1 such that d(ϕ(x), ϕ(y)) ≤ λ · d(x, y) for all x, y ∈ X. The constant λ is called a contractivity factor of ϕ.

Theorem 2.3.1 (Banach’s fixed point theorem) A contraction ϕ in a complete metric space has exactly one fixed point in X.

Proof. (i) By induction, one shows that for any pair x, y ∈ X

d(ϕ^{n}(x), ϕ^{n}(y)) ≤ λ^{n} · d(x, y) .

(ii) Using the triangle inequality, we get for all x ∈ X,

d(x, ϕ^{n}x) ≤ ∑_{k=0}^{n−1} d(ϕ^{k}x, ϕ^{k+1}x) ≤ ∑_{k=0}^{n−1} λ^{k}d(x, ϕ(x)) ≤ \frac{1}{1 − λ} · d(x_{0}, x_{1}) .

(iii) For all x ∈ X, x_{n} = ϕ^{n}(x) is a Cauchy sequence and has by completeness of X a limit x̃. Indeed, by (i),(ii),

d(x_{n}, x_{n+k}) ≤ λ^{n} · d(x, x_{k}) ≤ λ^{n} · \frac{1}{1 − λ} · d(x, x_{1}) .

(iv) The limit x̃ does not depend on x: assume there are two limits x̃, ỹ. They are both invariant under ϕ and satisfy

d(x̃, ỹ) = d(ϕ(x̃), ϕ(ỹ)) ≤ λ · d(x̃, ỹ) .

This is only possible if x̃ = ỹ. □

Corollary 2.3.2 Let M be a topological space. Given a family {ϕ_{t}}_{t∈M} : X → X of contractions with contractivity factor λ. Assume for all x ∈ X t ↦ ϕ_{t}(x) is continuous. Then, the fixed point x̃_{t} of ϕ_{t} is depending continuously on t.

Proof. The metric space C(M, X) of all continuous functions from M to X with metric δ(f, g) = sup_{t∈M} d(f(t), g(t)) is complete. The map Φ : f ↦ Φ(f)(t) = ϕ(f(t)) is a contraction with contractivity factor λ. It has therefore exactly one fixed point t ↦ x̃(t). Each x̃(t) ∈ X is a fixed point of ϕ_{t}. □

2.4 The implicit function theorem

Given a vector space X. A function || · || : X → [0, ∞) is called a norm, if d(x, y) := ||x − y|| is a metric and if for all x ∈ X and λ ∈ C ||λx|| = |λ| · ||x||. A normed vector space (X , || · ||) is a vector space, whith a norm || · || : X → [0, ∞). A Banach space (X , || · ||) is a complete normed vector space. We write B_{r}(x) := {y | ||y − x|| < r} for a ball of radius r around x ∈ X.

Examples:

(1) X = R^{d} with the Eucledean norm.

(2) Let X be a compact metric space and denote by C(X) the vector space of all continuous functions f : X → C. Take on C(X) the norm ||f|| = sup_{x∈X} |f(x)|.

(3) Given two Banach spaces X , Y, the set of linear maps A : X → Y which satisfy ||A|| = sup_{||x||=1} ||A(x)|| < ∞ form a Banach space.

Given two Banach spaces X , Y. A map F : X → X is differentiable at a point x ∈ X, if there exists a linear map DF (x) : X → Y such that for all y ∈ Y

lim_{ϵ→0} ||ϵy||^{−1} · ||F (x + ϵy) − F (x) − DF (x)ϵy|| = 0 .

In this case, the linear map DF (x) is called the (Fréchet) derivative of F at the point x ∈ X. If F is differentiable on an open set U and x ↦ DF (x) is a continuous map from U to B(X , Y), we say that F is continuously differentiable in U.

Examples.

(1) The map F : B(X , Y) → B(X , Y) given by F (A) = A^{−1} is differentiable on the open set of invertible A. The derivative is given by DF (A)U = −A^{−2}U.

(2) Take on (C(X), || · ||) the map F : C(X) → C(X) given by

F (f) = P (f(x))

where P is a differentiable function. Then DF (f)(g) = P′(f) · g.

Lemma 2.4.1 Let X = \overline{B_{r}(x_{0})} ⊂ X and assume ϕ is a differentiable map having the derivative

||Dϕ(x)|| ≤ λ < 1 for all x ∈ X. Then ϕ is a contraction in X = B_{r}(x_{0}).

Proof. Given two points x, y ∈ X. Connect x and y by a segment L(x, y) = {x+t(y−x) | t ∈ [0, 1]}. For each point w ∈ X, there exists an open neighborhood U(w) of x such that ||ϕ(z) − ϕ(w) − Dϕ(w)(z − x)|| ≤ ϵ(||z − w||) for all z ∈ U(w). This means for all z ∈ U(w)

||ϕ(z) − ϕ(w)|| ≤ (λ + ϵ) · ||z − w|| .

Cover L(x, y) by a countable set of balls U(w_{i}) such that in each ball this estimate holds. Summing up over the adjacent intervals U(w_{i}) ∩ L(x, y) gives

||ϕ(y) − ϕ(x)|| ≤ (λ + ϵ) · ||y − x|| .

Because ϵ was arbitrary, we have the contraction property. □

Corollary 2.4.2 Let X = \overline{B_{r}(x_{0})} ⊂ X and assume ϕ is a differentiable map X → X. If for all x ∈ X, ||Dϕ(x)|| ≤ |λ| < 1 and

||ϕ(x_{0}) − x_{0}|| ≤ (1 − λ) · r

then ϕ has exactly one fixed point in X.

Proof. The condition ||x − x_{0}|| < r implies that

||ϕ(x) − x_{0}|| ≤ ||ϕ(x) − ϕ(x_{0})|| + ||ϕ(x_{0}) − x_{0}|| ≤ λr + (1 − λ)r = r .

The map ϕ maps therefore the ball X into itself. Banach’s fixed point theorem applied to the complete metric space X and the contraction ϕ implies the result. □

Given three Banach spaces X , Y, Z and a differentiable map F : X ×Y → Z. We write D_{1}F and D_{2}F for the partial derivatives with respect to the first or second variable.

Theorem 2.4.3 (Implicit function theorem) Given a map F : X × Y → Y which is continuously differentiable on an open subset U ⊂ X × Y. Given a point (x_{0}, y_{0}) ∈ U with F (x_{0}, y_{0}) = 0, for which D_{2}F (x_{0}, y_{0}) is invertible. There exists a neighborhood V of x_{0} and a unique continuous map G : V → Y, such that G(x_{0}) = x_{0} and F (x, G(x)) = 0 for all x ∈ V.

Proof. The equation F (x, y) = 0 can be written as a fixed point equation y = ϕ_{x}(y) for the map

y ↦ ϕ_{x}(y) := y − L^{−1}F (x, y) ,

where L = D_{2}F (x_{0}, y_{0}). The value of G(x) is the fixed point y = ϕ_{x}(y).

We show that y ↦ ϕ_{x}(y) is a contraction on B_{r}(x_{0}) for x ∈ B_{R}(x_{0}) if R and r are small enough. First, we calculate

Dϕ_{x}(y) = 1 − L^{−1}D_{2}F (x, y) = 1 − L^{−1}(D_{2}F (x, y) − L) − L^{−1}L = −L^{−1}(D_{2}F (x, y) − L) .

Because (x, y) ↦ D_{2}F (x, y) is by assumption continuous in U and Dϕ_{x_{0}}(y_{0}) = 0, we get ||Dϕ_{x}(y)|| ≤ 1/2 for x in a ball B_{R}(x_{0}) ⊂ X and y in a ball X = B_{r}(y_{0}) ⊂ Y. For small enough R we acchieve also ||ϕ_{x}(y_{0})−y_{0}|| ≤ (1−λ)·r for all x ∈ B_{R}(x_{0}). Banach’s fixed point theorem 2.4.2 implies then that there is a unique fixed point G(x) of ϕ_{x}(G(x)) = G(x). This means F (x, G(x)) = 0. Corollary 2.3.2 implies that x ↦ G(x) is continuous. □

2.5 Embedding a shift in the Hénon map

We return now the Hénon map given by

H : (x, y) ↦ (x^{2} + c − by, x) ,

with real parameters b and c. If c < 0, we can introduce the new parameter a = 1/\sqrt{−c} and the new coordinates q = x · a, p = y · a to get the map

H̃ : (q, p) ↦ (\frac{q^{2} − 1}{a} − bp, q) .

The next theorem is a prototype for later results.

Theorem 2.5.1 (Devaney, Nitecki, 1979) Fix b ≠ 0. There exists a constant M > 0 such that for−c > M, the Hénon map H has a shift (X, T) with two symbols embedded.

Proof. The map

H̃ : ( \begin{matrix}q_{n} \\ q_{n−1} \end{matrix} ) ↦ ( \begin{matrix}\frac{q_{n}^{2}−1}{a} − bq_{n−1} \\ q_{n} \end{matrix} ) = ( \begin{matrix}q_{n+1} \\ q_{n} \end{matrix} )

can be written as

a · q_{n+1} + a · b · q_{n−1} = q_{n}^{2} − 1 .

We are now looking for sequences q_{n} solving this equation. We take sequences which are of the form q_{n} = q(T^{n}x), where (X = {−1, 1}^{Z}, T) is the shift, q ∈ C(X) and x ∈ X.

We rewrite the above equation as

a · q(T x) + a · b · q(T^{−1}x) − (q(x)^{2} − 1) = 0 .

With the map F : R × C(X) → C(X) defined by

F (a, q)(x) = a · q(T x) + a · b · q(T^{−1}x) − (q(x)^{2} − 1)

this equation can be written as F (a, q) = 0. The partial derivative D_{2}F (a, q) is given by

D_{2}F (a, q)u = a(u(T) + b · u(T^{−1})) − 2q · u .

(This is a discrete Schrödinger operator!). The map F (0, q) : C(X) → C(X) has the property that every function q ∈ C(X) with values in {−1, 1} is a solution of F (0, q) = 0. We take for such a solution the map q(x) = x_{0}. The derivative D_{2}F (0, q) is the diagonal operator

(D_{2}F (0, q)u = −2q · u

which is invertible. By the implicit function theorem, there exists a solution a ↦ q_{a} = G(a) satisfying F (a, q_{a}) = 0 for small enough a. We define now ϕ_{a} : X → R^{2} by

ϕ_{a}(x) = (q(x), q(T^{−1}x)) .

(i) This map ϕ_{a} is continuous, because q and T are continuous. (ii) Using F (a, q) = 0, we check that

ϕ_{a} ∘ T (x) = (q(T x), q(x))) = (\frac{(q(x)^{2} − 1)}{a} − b · q(T^{−1}x), q(x))
= H̃(q(x), q(T^{−1}x)) = H̃ ∘ ϕ_{a}(x)

for all x ∈ X. (iii) The map is injective because if two points x, y are maped into the same point in R^{2} then the fact that q_{a}(x) is near to q_{0}(x) = x_{0} implies that x_{0} = y_{0}. Because of the conjugation ϕ_{a} ∘ T^{n}(x) = H̃^{n} ∘ ϕ_{a}(x) we get also T^{n}(x) = T^{n}(y) and so x_{n} = y_{n} for all n. (iv) ϕ has a continuous inverse because every bijective map from a compact metric space to a metric space has a continuous inverse. The map is therefore a homeomorphism from X to a closed subset ϕ(X) ⊂ R^{2}. We finishes the proof that the shift is isomorphic to a subsystem of the Hénon map H̃ if a is small enough. In the old coordinates this corresponds to a shift embedded in H for −c large enough. □

2.6 Existence of the stable and unstable manifolds

Let T : R^{2} → R^{2} be a differentiable invertible map. If there exists x ∈ X and n ∈ Z such that T^{x} = x, then x is called a periodic orbit of T. The number n is called the period if x if it is the smallest amoung the numbers m satisfying T^{m}x = x. A periodic orbit of T with period n is a fixed point of T^{n}. Denote with λ, μ ∈ C the eigenvalues of the Jacobean dT^{n}(x). A periodic orbit x is called hyperbolic if |λ| ≠ 1, |μ| ≠ 1. A periodic orbit is called attracting, if |λ|, |μ| < 1, it is called repelling if |λ|, |μ| > 1.

Assume now T has a hyperbolic fixed point P. Assume the eigenvalues λ, μ of dT (P) satisfy by 0 < λ < 1 < μ. The stable manifold of P is defined as

W_{s}(P) = {(x, y) ∈ R^{2} | T^{n}(x, y) → P, n → ∞} .

The unstable manifold W_{u}(P) of P is defined to be the stable manifold of P with respect to the map T^{−1}. The local stable and unstable manifolds are the intersection of W_{s} rsp. W_{u} with a small neighborhood of P. From the local stable and unstable manifold, one obtains the global ones by taking the union of the images of the local ones.

The local stable and unstable manifolds are different on sufficiently small neighborhoods of P. However, the stable and unstable manifolds can globally be the same as the following example shows. Let T be the time-one map of the differential equation ẋ = y, ẏ = x − x^{3}. This differential equation can be written as one differential equation of second order ẍ−kx+ x^{3} = 0. If we multiply this equation with ẋ and integrate, we see that I = ẋ − kx^{2} + \frac{1}{2}x^{4} is constant. The sytem is integrable, has for k = 1 the stationary points (x, ẋ) = (0, 0) and (x, ẋ) = (±1, 0) from which the origin is hyperbolic. The stable and unstable manifold of (0, 0) is the level curve I = 0 and has the shape ∞.

Proposition 2.6.1 The stable and unstable manifolds through a hyperbolic fixed point P exist.

Proof. It is enough to prove the existence of invariant curves in a neighborhood of P. Because the unstable manifold is the stable manifold for T^{−1}, it is enough to prove the existence of the local stable manifold. After a linear coordinate transformation, the map T can be brought into a form

(x, y) ↦ (λx + f(x, y), μy + g(x, y))

in a neighborhood of the fixed point P = (0, 0). The maps f, g : R^{2} → R are smooth and satisfy f(0, 0) = dg(0, 0) = 0. Given ϵ > 0, there exists a neighborhood {|(x, y)| ≤ r} of P, such that the functions f, g satisfy |df(x, y)|, |dg(x, y)| ≤ ϵ. We will show that there exists a curve y = ϕ(x) with ϕ ∈ C([−r, r]) passing through P and which is T invariant. The condition for invariance is

μϕ(x) + g(x, ϕ(x)) = ϕ(λx + f(x, ϕ(x))) .

This condition can be rewritten as

ϕ(x) = μ^{−1}(ϕ(λx + f(x, ϕ(x))) − g(x, ϕ(x))) = F (ϕ(x))

which defines a map F. We show that for small enough ϵ the map F is a contraction in the space of all Lipschitz continuous functions in C[−r, r], where r is chosen such that |Df(x, y)| ≤ ϵ and |Dg(x, y)| ≤ ϵ for |x|, |y| ≤ r. Given h ∈ C([−r, r]). For ϵ < (μ − λ)/4r, the Lipschitz continuous functions are invariant:

|F (h)(x) − F (h)(x′))| ≤ μ^{−1}(λ|x − x′| + sup_{z∈[−r,r]} |Df(z, h(z))|2r|x − x′|
+ sup_{z∈[−r,r]} |Dg(z, h(z))|2r|x − x′|
≤ μ^{−1}(λ + 4ϵr)|x − x′|

Given h, k ∈ C([−r, r]). Wir show, that F is a contraction for ϵ < (μ − 1)/2.

|F (h)(x) − F (k)(x)| ≤ μ^{−1}(|h(λx + f(x, h(x)) − k(λx + f(x, k(x))|
+ |g(x, h(x)) − g(x, k(x))|)
≤ μ^{−1}(|h(λx + f(x, h(x)) − h(λx + f(x, k(x))|
+ |h(λx + f(x, k(x)) − k(λx + f(x, k(x))| + ϵ|h(x) − k(x)|
≤ μ^{−1}(|f(x, h(x)) − f(x, k(x))| + ϵ|h − k|_{∞} + ϵ|h − k|_{∞}
≤ μ^{−1}(1 + 2ϵ)|h − k|_{∞}

F leaves also the function satisfying h(0) = 0 invariant. Banach’s fixed point theorem assures now the existence of the stable manifold. □

2.7 A strange attractor?

In section, we consider the Hénon map H_{a,b}

( \begin{matrix}X \\ Y \end{matrix} ) ↦ ( \begin{matrix}Y + 1 − aX^{2} \\ bX \end{matrix} ) ,

with parameters a = 1.4, b = 0.3 as Hénon was studiing it. A coordinate transformation x = −aX, y = −aY, c = −a gives the map (x, y) ↦ (x^{2} + c − by, x) in the form as we saw it earlier. We concentrate here on experimental numerical investigations of the Hénon map.

Here, we see a sequence of snapshots zooming into the Hénon attractor

f[x_]:=N[1-1.4*x^2]; T[{x_Real,y_Real}]:=N[{y+f[x],0.3*x}];
S[p_]:=T[p];c=Last[NestList[S,{0.0,0.0},306]];F=NestList[S,c,10000];
G[R_]:=ListPlot[F,Frame->True,Axes->False,PlotRange->R];
Do[G[{{c[[1]]-2^k,c[[1]]+2^k},{c[[2]]-2^k,c[[2]]+2^k}}],{k,0,-10,-1}];

Basic observations:

• The map can be decomposed into a compression followed by a shear and a reflection:

(x, y) ↦ (x, by) ↦ (x, 1 − ax^{2} + by) ↦ (1 − ax^{2} + by, x) .

This allows a geometrical visualisation of the dynamics. A rectange is compressed then bended and stretched and reflected.

• For b = 0 the map reduces to the one-dimensional map x ↦ 1 − ax^{2} which is conjugated to the logistic map.

• It can be shown that the convex quadrilateral with vertices

P_{1} = (−1.33, 0.42), P_{2} = (1.32, 0.133), P_{3} = (1.245, −0.14), P_{4} = (−1.06, −0.5)

is maped into itself in the case q = 1.4, b = 0.3. It is called a trapping region. If a point reaches this set, it can’t leave it any more. The basin of attraction, the set of points in the plane which are finally in the trapping region is larger.

The fixed points of H_{a,b} are the solutions of 1 − ax^{2} + bx = x. For a > 0 there are two real roots (x_{±}, y_{±}) which can by lazy persons be computed with computer algebra:

Solve[1-a*x^2+b*x==x,x]

Putting in this in the Jacobi matrix dH_{a,b}(x, y) and calculating the eigenvalues can be done by

 Eigenvalues[{{-a*2x,1},{b,0.}} /. Solve[1-a*x^2+b*x==x,x][[2]]]

for i = 1, 2. Setting a = 1.4, b = 0.3 gives the eigenvalues 1.62678, −0.860594 at the first point and −2.20198, 0.635791 at the second point. We see that both points are hyperbolic. In such a case there exist two curves called stable and unstable manifold passing through the hyperbolic fixed point. These curves are invariant under the map and at the fixed point, the eigenvectors are tangent to these curves. In the next section we will give the precise definition of these manifolds and prove their existence.

The stable and unstable manifolds of a hyperbolic fixed point can be computed by a method of Francescini and Russo [35] We are looking for a curve in γ : R ↦ R^{2}, γ(t) = (x(t), y(t)) satisfying γ(0) = (x_{0}, y_{0}) and

H_{a,b}(x(t), y(t)) = (1 − ax(t)^{2} + by(t), x(t)) = (x(λt), y(λt))

for all t ∈ R, where λ is one of the eigenvalues. We see that x(t) = y(λt) so that it is enough to calculate x(t). Making the ansatz x(t) = ∑ x_{n}t^{n} gives

∑_{n=0}^{∞}[x_{n}(λ^{n} − bλ^{−n}) + a ∑_{j=0}^{n} x_{j}x_{n−j}]t^{n} = 1

which allows to calculate x_{n} recursively. x_{0} is given by the fixed point condition

x_{0} = \frac{1}{2a^{2}} (b − 1 + \sqrt{(1 − b)^{2} + 4a})

x_{1} is undetermined and we normalise it putting x_{1} = 1. The value of x_{n} can be computed recursively by

x_{n} = \frac{−a ∑_{k=1}^{n−1} x_{j}x_{n−j}}{(λ^{n} − bλ^{−n}) + 2ax_{0}} .

Here is a calculation of the stable manifold:

a=1.4;b=0.3; T[{x_,y_}]:={b*y+1-a*x^2,x};
x0=((b-1)+Sqrt[(1-b)^2+4a])/(2a); l=-a*x0-Sqrt[a^2 x0^2+b];
f[x_]:=Module[{n=Length[x]},

  Append[x,-a*Sum[x[[j+1]]*x[[n-j+1]],{j,n-1}]/((l^n-b*l^(-n))+2a x0)]];
xx=Module[{y={x0,1}},Do[y=f[y],{8}];y];
x[t_]:=Sum[xx[[k+1]]*t^k,{k,0,8}]; y[t_]:=x[t/l];
p[t_]:=Module[{n=7+Floor[Log[Abs[t]]/Log[Abs[l]]],q},
  q={x[t/l^n],y[t/l^n]};Do[q=T[q],{n}];q];
Stable=ParametricPlot[{p[t]},{t,-100.0,100.0}];

Similiarily one gets the unstable manifold by replacing T with T^{−1}. Every point in the intersection of the stable and unstable manifold is called a homoclinic point.

The dimension of the attractor measures the fractal property. For the Hénon map one measures that the attractor is a fractal. There is however no proof of this. The Lyapunov exponent measures chaotic property of an attractor.

Given a point (x, y) in the plane. If the limit

lim_{n→∞} \frac{1}{n} log ||dH_{a,b}^{n}(x, y)||

exists, one calls its value the (upper) Lyapunov exponent of the map. Oseledec’s ergodic theorem says that if m is an invariant probability measure of H_{a,b}, almost all points have a Lyapunov exponent. If the Lyapunov exponent is positive, the measure theoretical dynamical system (R^{2}, H, m) is nonuniformly hyperbolic. It is in general a difficult problem to decide if there is such an invariant measure. For the Hénon map with the parameter values of Hénon, the Lyapunov exponent give an experimental value of 0.420809. We will see now how one gets such a measurement.

The following Mathematica program can be used to calculate the Lyapunov exponent: to see the value for a = 1.4 with 10 orbits of length 5000, one enters

Lyapunov[1.4,5000,10]

T[{x_,y_},a_]:={y+1-a*x^2,0.3x};
T[{x_,y_},a_,n_]:=Module[{p={x,y}},Do[p=T[p,a],{n}];p];
A[{x_,y_},a_]:=N[{{-a*2x,1},{0.3,0.0}}];
Lya[{x_,y_},a_,n_]:=Module[{t=0,B=IdentityMatrix[2],p=T[{x,y},a,100]},

  Do[B=A[p,a].B;p=T[p,a]; t=t+Re[Log[B[[1,1]]]];B=B/B[[1,1]],{i,n}];t/n];
Lyapunov[a_,n_,m_]:=Module[{R:=Random[]/10},Sum[Lya[{R,R},a,n],{m}]/m];

We measure that the Lyapunov exponent for increasing values of a:

Do[Print[Lyapunov[0.4+k*0.01,1000,2]],{k,0,100}]

2.8 Epilog

Why are Hénon type maps interesting?

• The Hénon map is the simplest twist map on the plane. Every quadratic map with constant Jacobian can be brought into the form of the Hénon map with two parameters.

• A Hénon map can be found in some limit as subsystems of arbitrary smooth maps near homoclinic tangencies.

• The conservative Hénon map is a prototype of a discrete Hamiltonian system on a noncompact phase space.

• By changing of parameters, there can occur universal phase transitions like accumulations of period doubling.

• For some parameters, it is proven now that a ”strange attractor” exists and that there is an invariant Gibbs measure.

A little bit of history:

• Hénon invents the Hénon map in 1976 [30]. As a physicist and astronomer he was motivated by differential equations coming from the motion of stars around galaxies or the Lorenz system. Hénon’s paper was rejected first (the referee is known to be Palis) because it was believed that the observed attractor is just a highly periodic orbit. The concept of ”strange attractor” was not so easy to accept 20 years ago. Even if this is up to now not yet decided weather the Hénon map has a strange attractor for the specific values chosen by Hénon, it turned out that the map is very interesting from the mathematical point of view. The map was studiied in the last decade by many people. We mention among them Marotto [41], Curryu [12], Milnor [46], Devaney-Nitecki [15], Benedicks-Young [5],

A short view on newer results:

Katok-Mendoza-Yomdim-Devaney-Nitecki: The topological entropy of the Hénon map H_{a,b} depends continuously on the parameters and takes all values in [0, 2]. Benedicks-Carlson (1991): There exists a set Y of positive Lebesgue measure in the parameter plane of the Hénon map H_{a,b} such that for (a, b) ∈ Y the map H_{a,b} has a topologically transitive attractor Λ_{a,b} on which there is some hyperbolic behavior. Benedicks-Young (1993): For (a, b) ∈ Y, the map H_{a,b} admits a unique SRB measure with support Λ_{a,b}. The measure theoretical dynamical system (Λ_{a,b}, H_{a,b}, λ) is Bernoulli. (An invariant measure is called a SBR measure, if the Lyapunov exponents are positive almost everywhere and the conditional measures of μ on the unstable manifolds are absolutely continuous with respect to the Lebesgue measure induced on these manifolds.)

2.9 Open research problems

Exercices:

• Show that the twist map given by x^{4} + c has for large enough c a Bernoulli shift embedded. What Bernoulli shifts can be embedded when the map is considered to be a map on C^{2}?

• A twist map T in the plane is integrable, if there exists a continuous function F : R^{2} → R such that F (T (x, y)) = F (x, y) and F^{−1}(a) is one dimensional. Show that the Colin-de Verdiere map is integrable. Hint: Show that T^{9} = Id. Show that if T^{n} = Id and F_{0}(x, y) = y then F (x, y) = ∑_{i=0}^{n−1} F (T^{n}(x, y)) is an integral.

F (x, y) = y + |y − |x|| + |x − |y − |x||| + |y − |x − |y||| + |x − |y| + |y − |x − |y|||| .

Open research problems:

• Is the Cohen-Verdiere map integrable for ϵ > 0?

• Is there are dense set of parameter values (a, b) for which the Hénon attractor is periodic?

2.10 Exercices

Topic: The shift 1) Reminder: We have defined the sequence space

X = {x = (…, x_{−2}, x_{−1}, x_{0}, x_{1}, x_{2} …) | x_{i} ∈ I} = I^{Z} .

where I is a finite set called alphabet. The shift was the map T : X → X, (T x)_{n} = x_{n+1}. We have defined d(x, y) = 2^{−N}, where N is the largest non-negative integer such that x_{i} = y_{i} for all |i| < N.

Topology of the space X

a) Show that d is a metric on X.

b) Show that the metric space (X, d) is compact.

c) Show that the shift map T : X → X is a homeomorphism of X.

2) Reminder: A topological dynamical system is said to have sensitive dependence on initial conditions, if there exists δ > 0 such that for all x ∈ X, ϵ > 0, there exists y ∈ X with d(x, y) < ϵ and n ≥ 1 with d(T^{n}x, T^{n}y) ≥ δ. A topological dynamical system is called transitive, if for any two open sets U, V ⊂ X, there exists n > 0, such that T^{n}(U) ∩ V is not empty.

Chaotic properties of the shift:

a) Prove that the shift has sensitive dependence on initial conditions.

b) Prove that the shift is transitive.

c) Prove that periodic orbits are dense.

A topological dynamical system with these three properties is called chaotic in the sense of Devaney.

Topics: Embedding of shifts, Integrability and nonintegrability of twist maps

1) Convince yourself that the twist map given by V′(x) = x^{2n} + c has for large enough c a shift embedded. Just indicate the changes, which have to be done in the proof of the course. What shifts can be embedded when the map is extended to an analytic map on C^{2}?

Reminder: A homeomorphism T : R^{2} → R^{2} is called S-integrable, if it is isomorphic to a map S : R^{2} → R^{2} given in polar coordinates by S : (r, ϕ) ↦ (r, ϕ + α(r)), where α(r) is a continuous function [0, R) ↦ R.

2) Show that there are parameter values such that the twist map H : (x, y) ↦ (x^{2n}+c−by, x) is not S-integrable.

Hint: Show that S can not have a shift embedded.

Advanced problem: (this can take more time! I don’t expect that you solve it. Think about it if you have free time).

3) Show that the Colin-de Verdiere map T (x, y) = (\sqrt{a^{2} + x^{2}} − y, x) is S-integrable for a = 0. It is then given by the map (x, y) ↦ (|x| − y, x).

Hints: Try to find a continuous function F : R^{2} → R such that F (T (x, y)) = F (x, y) (such an F is called an integral) and map the level curves F^{−1}(r) to the circles {(x, y) ∈ R^{2} | |(x, y)| = 1}.

To find F, show first that T^{9}(x, y) = (x, y) for all (x, y) ∈ R^{2}. Prove then a lemma saying that any such map is integrable: if there exists n such that T^{n}(x, y) = (x, y) for all (x, y) ∈ R^{2} and if g : R^{2} → R is any function, then F (x, y) = ∑_{i=0}^{n−1} g(T^{i}(x, y)) is an integral of T. Find a suitable g so that the level curves of g can be maped to the invariant circles of S. Conjugate the dynamics on each level curve of F to the dynamics of S on the circles.

Open research problems: (This can take infinite time!)

4) Is the Cohen-Verdiere map S-integrable for all a? Computer experiments support this conjecture.

5) How looks the set of parameter values (a, b) for which the Hénon attractor is a finite set.

Fractals

3.1 Hausdorff dimension and fractals

Let (X, d) be a metric space. Denote by

|A| = sup_{x,y∈A} d(x, y)

the diameter of a subset A. Define for ϵ > 0, s > 0

h_{ϵ}^{s}(A) = \begin{matrix}inf \\ U & ϵ \end{matrix} ∑_{U∈U_{ϵ}} |U|^{s} ,

where U_{ϵ} runs over all countable open covers of A with diameter < ϵ. Such covers are also called ϵ-covers. The limit

h^{s}(A) = lim_{ϵ→0} h_{ϵ}^{s}(A)

is called the s− dimensional Hausdorff measure of the set A. Note that the limit exists in [0, ∞], because ϵ ↦ h_{ϵ}^{s}(A) is increasing for ϵ → 0.

Lemma 3.1.1

If h^{s}(A) < ∞, then h^{t}(A) = 0 for all t > s.

Proof. Take ϵ > 0 and assume {U_{j}}_{j∈N} is an open ϵ-cover of A. Then

h_{ϵ}^{t}(A) ≤ ∑_{j} |U_{j}|^{t} ≤ ϵ^{t−s} · ∑_{j} |U_{j}|^{s} .

Taking the infimum over all coverings gives

h_{ϵ}^{t}(A) ≤ ϵ^{t−s} · h_{ϵ}^{s}(A) .

In the limit ϵ → 0, we obtain from h^{s}(A) < ∞ that h^{t}(A) = 0. □

Corollary 3.1.2 Either there exists dim_{H}(A) ≥ 0 such that

s < dim_{H}(A) ⇒ h^{s}(A) = ∞ ,
s > dim_{H}(A) ⇒ h^{s}(A) = 0

or for all s ≥ 0, h^{s}(A) = 0. In the later case, one defines dim_{H}(A) = ∞.

The number dim_{H}(A) ∈ [0, ∞] is called Hausdorff dimension of A. A fractal is a subset of a metric space which has finite non-integer Hausdorff dimension.

Remarks.

1) The definition of a fractal is not canonical. An other definition is that a fractal is a set for which the topological dimension is strictly smaller than the Hausdorff dimension. The topological dimension dim_{T} (X) is defined inductively. If X = ∅, then dim_{T} (X) = −1. A topological space (X, U) has has zero dimension dim_{T} (X) = 0, if there exists a basis for U which consists only on clopen (=open and closed) sets. Inductively dim_{T} (X) ≤ k if there exists a basis V for U such that every V ∈ V satisfies dim_{T} (δV) ≤ (k − 1). The smallest k satisfying dim_{T} (X) ≤ k is the topological dimension of X. If for no k ∈ N dim_{T} (X) ≤ k, one writes dim_{T} (X) = ∞. It is known that

dim_{T} (X) = inf(dim_{H}(Y), Y homeomorphic X) .

2) In the definition of the Hausdorff dimension, one can restrict to coverings by balls because any set of diameter ϵ can be covered by a ball of radius 2ϵ. This restriction to balls changes the value of the s− dimensional Hausdorff measure only by a constant 2^{s}. This does not affect the dimension.

3) For compact sets, any open cover has a finite subcover. When measuring the Hausdorff dimension of a compact set A, one can restrict therefore to finite covers.

3.2 Properties of the Hausdorff dimension

We discuss now some properties of the Hausdorff dimension:

Lemma 3.2.1

A ⊂ B ⇒ dim_{H}(A) ≤ dim_{H}(B).

Proof. Every ϵ-cover of B is also an ϵ-cover of A. It follows that h^{s}(A) ≤ h^{s}(B). If h^{s}(B) = 0, then also h^{s}(A) = 0. Therefore, dim_{H}(A) ≤ dim_{H}(B). □

Lemma 3.2.2

dim_{H}(A ∪ B) ≤ max(dim_{H}(A), dim_{H}(B)).

Proof. Assume s > max(dim_{H}(A), dim_{H}(B)). Then s > dim_{H}(A) and so h^{s}(A) = 0. In the same way h^{s}(B) = 0. Because h^{s}(A∪B) ≤ h^{s}(A)+h^{s}(B) = 0, we get also dim_{H}(A∪B) ≤ s. □

Denote with B(x, r) the ball of radius r around x ∈ X.

Lemma 3.2.3 Assume A is a countable subset of X. Then dim_{H}(A) = 0.

Proof. Assume A = {a_{1}, a_{2}, …}. Given s > 0. Define the cover U_{n} = B(a_{n}, ϵ^{n/s}2^{−1}(n!)^{−1/s}). Because

∑_{n=1}^{∞} |U_{n}|^{s} = ∑_{n=1}^{∞} \frac{ϵ^{n}}{n!} = e^{ϵ} − 1

we get h^{s}(A) = lim_{ϵ→0} h_{ϵ}^{s}(A) = 0. Because s > 0 was arbitrary, we have dim_{H}(A) = 0. □

The following lemma is a useful general method to obtain a lower bound on the Hausdorff dimension:

Lemma 3.2.4 Suppose μ is a probability measure on X such that μ(A) = 1. Assume, there are constants c, R and t such that

μ(B(x, r)) ≤ c · r^{t},

for all 0 < r < R and x ∈ A. Then dim(A) ≥ t.

Proof. Take ϵ < R/2. Consider a cover of A by balls B(x_{k}, r_{k}) with r_{k} < ϵ. This ϵ-cover can be used to construct a 2ϵ-cover B(y_{k}, 2r_{k}) which satisfies y_{k} ∈ A. From

1 = μ(A) ≤ ∑_{j} μ(B(y_{j}, 2r_{j})) ≤ c · ∑_{j}(2r_{j})^{t} ≤ c · ∑_{j} |B(x_{k}, r_{k})|^{t} ,

it follows that h_{ϵ}^{t}(A) ≥ 1/c. For ϵ → 0, we get h^{t}(A) ≥ 1/c and therefore dim(A) ≥ t. □

Remark. The upper and lower t-dimensional density of a probability measure μ are functions on X defined as

\overline{θ}^{t}(μ, x) = lim sup_{r→0} μ(B(x, r))/(2r)^{t},
\underline{θ}^{t}(μ, x) = lim inf_{r→0} μ(B(x, r))/(2r)^{t} .

The above lemma tells that dim_{H}(A) ≥ t if the upper t- dimensional density of a probability measure on A is bounded by a constant.

Proposition 3.2.5 dim_{H}(R^{n}) = n .

Proof. Look at the cube Q = [0, 1]^{n} ⊂ R^{n}. It is covered by k^{n} small cubes with side 1/k and diameter \sqrt{n}/k. So, if ϵ ≥ \sqrt{n}/k, then

h_{ϵ}^{n}(A) ≤ k^{n} · (\sqrt{n}/k)^{n} = \sqrt{n}^{n} .

Therefore h^{n}(A) ≤ \sqrt{n}^{n} and dim_{H}(Q) ≤ n. On the other hand, the Lebesgue measure μ on Q has the property that

μ(B(x, r)) ≤ c · r^{n},

for some constant c depending only on the dimension. The above lemma implies that dim_{H}(Q) ≥ n. Because Q ⊂ R^{n}, we get dim_{H}(R^{n}) ≥ n. Given s > n, then h^{s}(Q) = 0. The space R^{n} can be covered by a countable set of translates Q_{k} of Q. We have h^{s}(R^{n}) ≤ ∑_{k} h^{s}(Q_{k}) = 0. This shows that dim_{H}(R^{n}) ≤ s. Since s > n was arbitrary, we have also dim_{H}(R^{n}) ≤ n. □

Corollary 3.2.6

If A ⊂ R^{n} contains an open ball B, then dim_{H}(A) = n.

Proof. Write A = Q ∪ A \ Q where Q is a cube of positive diameter in the ball B. Because dim_{H}(Q) = n, we have also dim_{H}(A) = n. □

Given two metric spaces X, Y. A map ϕ : X → Y is called Lipschitz continuous, if there exists a constant λ (called Lipschitz constant) such that

d(ϕ(x), ϕ(y)) ≤ λ · d(x, y), ∀x, y ∈ X .

A map ϕ : X → Y has bounded distortion if there exist two constants λ, μ > 0 such that

μ · d(x, y) ≤ d(ϕ(x), ϕ(y)) ≤ λ · d(x, y), ∀x, y ∈ X .

A map with bounded distortion is also called a metric isomorphism or a lipeomorphism. Note that a metric isomorphism from A to ϕ(A) is a bijection because ϕ must be injective. It follows that such a map is a homeomorphism onto its image and the inverse is Lipschitz continuous.

Lemma 3.2.7 Given a Lipschitz continuous map ϕ : X → Y, then dim_{H}(ϕ(A)) ≤ dim_{H}(A), for any set A ⊂ X. If ϕ : X → Y is a metric isomorphism, then dim_{H}(ϕ(A)) = dim_{H}(A).

Proof. A Lipschitz continuous map ϕ maps an ϵ-cover of A into a (λ · ϵ)-cover of ϕ(A). It follows that h^{s}(ϕ(A)) ≤ λ^{s} · h^{s}(A) and the claim follows. If ϕ is a metric isomorphism, then ϕ^{−1} is a Lipschitz map from ϕ(A) to A. It follows from the previous step that dim_{H}(A) ≤ dim_{H}(ϕ(A)). □

3.3 The Hutchinson operator

Let (X, d) be a complete metric space. Define for x ∈ X and A ⊂ X

d(x, A) = inf_{a∈A} d(x, a) .

A set A ⊂ X is called bounded, if |A| < ∞. Denote with B the space of closed bounded nonempty subsets of X. Define on B the Hausdorff distance

d(A, B) = inf_{ϵ>0}{ A ⊂ B_{ϵ}, B ⊂ A_{ϵ}} ,

where A_{ϵ} = {x ∈ X |d(x, A) ≤ ϵ} is called the ϵ-collar of A.

Lemma 3.3.1

(B, d) is a metric space.

Proof. (i) The symmetry d(A, B) = d(B, A) is clear from the definition. (ii) d(A, B) = 0 ⇒ A = B. The assumption d(A, B) = 0 implies that A is in the intersection of all ϵ-collars of B. Because B is closed, this intersection coincides with B. This implies A ⊂ B. In the same way, one shows that B ⊂ A. (iii) The triangle inequality d(A, C) ≤ d(A, B)+d(B, C) can be checked as follows: A is in a d(A, B)-collar of B and B is in a d(B, C)-collar of C, therefore A is in a (d(A, B)+d(B, C))- collar of C. A symmetric argument gives that C is in a (d(C, B) + d(B, A))-collar of A. □

Remark. If d′(A, B) = sup_{a∈A} d(a, B) is the distance of A and B, then the maximum between d′(A, B) and d′(B, A) is also the Hausdorff distance.

Lemma 3.3.2

The space (B, d) is complete.

Proof. (See also [22] p. 37 in the case X = R^{n} and [19] p. 67 in the general case.) Take a Cauchy sequence A_{m} in B.

(i) The set A = ⋂_{n} B_{n} with B_{n} = \overline{⋃_{m≥n} A_{m}} is closed as an intersection of a sequence of closed sets B_{n}. (ii) A is not empty: the sequence B_{n} is nested: B_{m} ⊂ B_{n} for m ≥ n. If |A| > 0 then A is not empty. Assume therefore |A| = 0. Then |B_{n}| → 0. Any sequence x_{n} satisfying x_{n} ∈ B_{n} is a Cauchy sequence. By completeness, it has a limit x ∈ A and A is not empty. (iii) ⋃_{m} A_{m} is bounded: take ϵ > 0 and N such that d(A_{k}, A_{l}) ≤ ϵ for all k, l > N. Then ⋃_{m}(A_{m}) is a subset of the ϵ-collar of the bounded set ⋃_{m=1}^{N} A_{m} and A is bounded. (iv) In order to show that A_{n} converges to A it suffices to show that d(B_{n}, A_{n}) → 0 because B_{n} → A. Given ϵ > 0. Take N so large that d(A_{k}, A_{l}) ≤ ϵ for all k, l > N. Because d(A_{N}, A_{k}) ≤ ϵ for all k ≥ N, we have also d(A_{N}, B_{N}) ≤ ϵ. □

Lemma 3.3.3 If X is compact, then C, the subset of compact elements in B is compact in B.

Proof. A closed set Y ⊂ X is compact if and only if we can find for each ϵ > 0 a finite set K(ϵ) = {x_{1}, … x_{N}(ϵ)} such that Y ⊂ ⋃_{n=1}^{N(ϵ)} B(x_{n}, ϵ). In order to show that C is compact, we construct for each ϵ > 0 the finite (and so compact) set K(ϵ) = {L | L ⊂ K(ϵ)} and prove compactness of C by showing that for each Y ∈ C, there exists K ∈ K(ϵ) such that d(Y, K) ≤ ϵ: given Y ∈ C. Define

K(Y) = {x_{j} ∈ K(ϵ) | B(x_{j}, ϵ) ∩ Y ≠ ∅} .

By construction Y ⊂ K(Y)_{ϵ}. On the other hand, any x ∈ K(Y) satisfies d(x, Y) ≤ ϵ and so K(Y) ⊂ Y_{ϵ}. We have shown that d(K(Y), Y) ≤ ϵ. □

An iterated function system is a finite set of continuous contractions f_{1}, f_{2}, …, f_{n} on X.

Given an iterated function system f_{1}, f_{2}, …, f_{n}. The Hutchinson operator F is a map on C defined as the map

F : C → C, F (A) = ⋃_{j=1}^{n} f_{j}(A) .

Lemma 3.3.4

For A, B, C, D ∈ B we have

d(A ∪ C, B ∪ D) ≤ max{d(A, B), d(C, D)} .

Proof. Assume without loss of generality d(A, B) ≥ d(C, D). A is in a d(A, B)-colar of B and so in a d(A, B)-colar of B∪D. Also C is in a d(C, D)-colar of D and so in a d(A, B)-colar of D and so in a d(A, B)-colar of B ∪ D. Together, A is in a d(A, B)-colar of B ∪ D. In the same way, C is in a d(A, B)-colar of B ∪ D and A ∪ C is in a d(A, B)- colar of B ∪ D. The same argumentation chain with (A, B) and (C, D) interchanged gives that B ∪ D is in a d(A, B)-colar of A ∪ B. □

Proposition 3.3.5 The map F is a contraction on C. The contraction rate is λ = max_{j} λ_{j}, where λ_{j} are the contraction rates of f_{j}.

Proof. We prove by induction that for any two finite collection of sets {A_{j}}_{j=1}^{m}, {B_{j}}j = 1^{m}

d( ⋃_{j=1}^{m} A_{j}, ⋃_{j=1}^{m} B_{j}) ≤ max_{j} d(A_{j}, B_{j}) .

For m = 1 it is trivial. The induction step is proven using Lemma 3.3.4.

Applying this to A_{j} = f_{j}(A), B_{j} = f_{j}(B) and using d(f_{j}(A), f_{j}(B)) ≤ λ · d(A, B) gives d(F (A), F (B)) ≤ λ · d(A, B). □

Remark. It follows immediatly from the contraction property that F is Lipschitz continuous and so continuous.

Because a contraction in a complete metric space has a unique fixed point, there exists a fixed point of F, which is called the attractor of the iterated function system.

Iterated function systems are often acting in the metric space R^{n}. An affine map A_{b} on R^{n} is a linear transformation A followed by a translation x ↦ x + b:

A_{b} : x ↦ Ax + b .

Because computers can work very well with affine maps, one considers often iterated function systems, where the contractions f_{i} are affine maps. In the next section we see some examples.

3.4 Examples

We give here a list of the most famous fractals.

1) The Cantor set. Define on R the iterated function system f_{1}(x) = 1/3 · x, f_{2}(x) = 1/3 · x + 2/3. The Cantor set is the attractor of f_{1}, f_{2}. The dimension is log(2)/ log(3) as we have seen in the exercice. Replacing n = 2, m = 3 by any other pair 2 ≤ m < n gives Cantor sets with dimension log(m)/ log(n).

2) The Koch curve. Given the points

P_{1} = (0, 0), P_{2} = (1/3, 0), P_{3} = (1/2, \sqrt{3}/6), P_{4} = (2/3, 0), P_{5} = (1, 0) .

Define f_{j} the unique similitude mapping P_{1}P_{5} to P_{i}P_{i+1} and having positive determinant. The attractor of f_{1}, f_{2}, … f_{5} is called the Koch curve. The contraction rate of f_{j} is 1/3. The dimension is therefore log(5)/ log(3).

3) The Koch snowflake or Koch iland is obtained by replacing the edges of a regular triangle by a Koch curve. Three dimensional analogues with polyhedra can be computed easily with Mathematica

<<Graphics`Polyhedra`
Fractal[0]:=Dodecahedron[]; Fractal[n_]:=Stellate[Fractal[n-1]];
Do[Show[Graphics3D[Fractal[k]]],{k,1,4}]

4) The Sierpinski gasket. Define on R^{2} the iterated function system f_{1}(x) = 1/2x + (1/4, 0), f_{2}(x) = 1/2x − (1/4, 0), f_{3}(x) = 1/2x + (0, \sqrt{3}/2). The attractor is the Sierpinski gasket with dimension log(3)/ log(2).

5) The Sierpinski carpet. Define on R^{2} the iterated function system f_{i}(x) = 1/3x+v_{i}, where v_{i} are one of the 8 vectors (j/3, k/3), 0 ≤ j, k ≤ 2 with (j, k) ≠ (1, 1). The attractor is the Sierpinski carpet with dimension log(8)/ log(3). It is an interesting object because Sierpinski showed that any compact curve in the plane is topologically homeomorphic to a compact subset of the Sierpinski carpet. In contrast, the Sierpinski gasket has not this property.

Shirp[0,{i1_,j1_}]:={Rectangle[{i1,j1}]};
Shirp[n_,{i2_,j2_}]:=Module[{s={}},Do[If[Not[u==v==2],
 s=Union[s,Shirp[n-1,{u*3^(n-1)+i2,v*3^(n-1)+j2}]]],{u,3},{v,3}];s];
Show[Graphics[Shirp[4,{0,0}]]];

f={Function[x,{{1/3,0},{0,1/3}}.x          ],
   Function[x,{{1/3,0},{0,1/3}}.x+{1/3,0. }],
   Function[x,{{1/3,0},{0,1/3}}.x+{1/3,2/3}],
   Function[x,{{1/3,0},{0,1/3}}.x+{2/3,1/3}],
   Function[x,{{1/3,0},{0,1/3}}.x+{2/3,0. }],
   Function[x,{{1/3,0},{0,1/3}}.x+{0. ,1/3}],

   Function[x,{{1/3,0},{0,1/3}}.x+{0.,2/3 }],
   Function[x,{{1/3,0},{0,1/3}}.x+{2/3,2/3}]}
g[x_]:=N[f[[Random[Integer,7]+1]][x]]; start:=Last[NestList[g,{0.,0.},20]];
ListPlot[NestList[g,start,10000], Axes->False];

6) The Menger sponge. Define on R^{3} the iterated functon system f_{i}(x) = 1/3x + v_{i}, where v_{i} are one of the 20 vectors (j/3, k/3, l/3). Where (i, j, z) is running over all tripels 0 ≤ j, k, l ≤ 2 not satisfying i = j = 1 or i = k = 1 or j = k = 1. The attractor has dimension log(20)/ log(3). The construction can be generalized in any dimension R^{n}. For n = 3, Menger showed that any one-dimensional compact curve in R^{3} is homeomorphic to a compact subset of the Menger sponge.

<<Graphics`Polyhedra`
Menger[0,{i_,j_,k_}]:={Cuboid[{i,j,k}]};
Menger[n_,{i_,j_,k_}]:=Module[{s={}},Do[If[Not[u==v==2 || v==w==2 || u==w==2],
 s=Union[s,Menger[n-1,3^(n-1)*{u,v,w}+{i,j,k}]]],{u,3},{v,3},{w,3}];s];
Show[Graphics3D[Menger[2,{0,0,0}]]];

7) Julia sets. This is an example of an attractor of a nonaffine iterated function system. Define on C = R^{2} the iterated function system f_{1}(z) = +\sqrt{z − E}, f_{2}(z) = −\sqrt{z − E}. For large E, these maps are contractions on a ball B(r) with radius \sqrt{E}, because |f′_{i}(z)| = \frac{1}{2\sqrt{z−E}}. Similarily, one can replace the square root by the n− th square root and get like this attractors of an interated function system with n functions.

c=1.0+1.0*I;J[0,z_,r_]:={Disk[{Re[z],Im[z]},r]};
J[n_,z_,r_]:=Union[J[n-1,Sqrt[z-c],r/2^(n-1)],J[n-1,-Sqrt[z-c],r/2^(n-1)]];
Do[Show[Graphics[J[n,0+0*I,1]]],{n,0,5}];

8) Barnsleys fern.

f={Function[x,{{ 0.000, 0.000}, { 0.000,0.270}}.x+{ .5  ,      0}],
   Function[x,{{-0.139, 0.263}, { 0.246,0.224}}.x+{ 0.57, -0.036}],
   Function[x,{{ 0.170,-0.215}, { 0.222,0.176}}.x+{ 0.408, 0.089}],
   Function[x,{{ 0.781, 0.034}, {-0.032,0.739}}.x+{ 0.1075,0.270}]};
U:=Module[{u=Random[]},If[u<0.02,1,If[u<0.17,2,If[u<0.3,3,4]]]];
g[x_]:=f[[U]][x]; start:=Last[NestList[g,{0,0},100]];
ListPlot[NestList[g,start,10000], Axes->False];

9) The Solenoid. This is an attractor of a nonaffine iterated function system which has only one generator. The basin of attractor is a solid torus. In a cylindrical coordinates, the attractor is obtained by an affine iterated function system.

T[{z_,w_}]:={z^2,z/2+w/4};  T[n_,{z_,w_}]:=Last[NestList[T,{z,w},n]];
S[{z_,w_}]:={Re[z](1+Re[w]/4),Im[z](1+Re[w]/4),Im[w]};
ParametricPlot3D[S[T[2,{Exp[I*t],Exp[I*s]}]],{s,0,N[2Pi]},{t,0,N[2Pi]},
  PlotPoints->100];
Table[ParametricPlot3D[S[T[n,{Exp[I*t],0.}]],{t,0,N[2Pi]},PlotPoints->700],{n,4}];

10) The Antoine Necklace. Take a solid torus A and find n affine maps f_{i} which map A into itself and such that f_{i}(A) is cyclically linked with f_{i+1}(A) and f_{i−1}(A) and with no other images. The attractor is the Antoine Necklace. It is totally disconnected but no embedded 2 sphere in R^{3} is able to seperate it. The iterated function system can be given more precicely. Take the circle S = (cos(t), sin(t), 0) and let A be the solid torus which is a 1/4-collar of S. Take the iterated function system f_{j}(x) = R_{j}(x/6 + v_{j}) where v_{j} = (cos(2πj/12), sin(2πj/12), 0)) and R_{j} is a rotation with angle j · π/2 about the axes {v_{j} + t(− sin(2πj/12), cos(2πj/12), 0) | t ∈ R}.

3.5 The box counting dimension

Given a compact set A ⊂ X. Define for ϵ > 0, N_{ϵ}(A) as the smallest number of sets of diameter ϵ which cover A. By compactness of A, this is a finite number. Define the upper capacity

\overline{dim}_{B}(A) = lim sup_{ϵ→0} \frac{log(N_{ϵ}(A))}{− log(ϵ)}

and analoguous the lower capacity \underline{dim}_{B}(A), where lim sup is replaced with lim inf. If the lower and upper capacities coincide, the value dim_{B}(A) is called box counting dimension of A. It is also called fractal dimension.

Remarks:

1) The Hausdorff dimension is difficult to calculate numerically. The central difficulty is to determine the infimum over ∑_{i} |U_{i}|^{t}, where U = {U_{i}} is an ϵ-cover of A. The box-counting dimension simplifies this problem by replacing the terms |U_{i}|^{t} by ϵ^{t}. The prize one has to pay is that one can no more measure all the bounded sets like this. In general, the upper and lower capacities differ. The name ”box counting dimension” comes from the numerical method to compute this number for subsets of R^{n}. Cover R^{n} by closed square boxes of side length 2^{−k}. and let M_{k}(A) be the number of such boxes which intersect A.

Proposition 3.5.1 Assume we are in R^{n}. If the upper and lower capacity coincide then

dim_{B}(A) = lim_{k→∞} \frac{log(M_{k}(A))}{log(2^{k})} .

Proof. Any set of diameter 2^{−k} can intersect at most 2^{n} grid boxes. On the other hand, any box of side 2^{−k} has diameter smaller than 2^{−k+1}. There exists therefore a constant C only depending on the dimension n such that

C^{−1} · M_{k}(A) ≤ N_{2^{−k}}(A) ≤ C · M_{k}(A) .

Therefore

lim_{k→∞} \frac{log(M_{k}(A))}{log(2^{k})} = lim_{k→∞} \frac{log(N_{2^{−k}}(A))}{log(2^{k})} .

□

3.6 The similarity dimension

The dimension of an attractor of special affine iterated function systems can often be calculated. To do so, the maps f_{j} must be of a special form.

A transformation ϕ of a metric space (X, d) is called a similarity if there exists λ > 0, such that d(ϕ(x), ϕ(y)) = λ · d(ϕ(x), ϕ(y)). A similarity with λ = 1 is called an isometry. By definition, every similarity is also a metric isomorphism.

Proposition 3.6.1 (Barnsley) Assume A is an attractor of an iterated function system generated by n contracting similarities f_{i} having contraction rates λ_{j}. Assume A_{j} = f_{j}(A) are all pairwise disjoint and assume the box counting dimension dim_{B}(A) of A exists. Then dim_{B}(A) is the unique solution d of ∑_{j} λ_{j}^{d} = 1.

Proof. For sufficiently small ϵ, a set of diameter ϵ can not intersect two different sets A_{j}. We get therefore for such ϵ

N_{ϵ}(A) = ∑_{j} N_{ϵ}(A_{j}) .

Because by assumption N_{ϵ}(A_{j}) = N_{λ^{−1}ϵ}(A), we get

N_{ϵ}(A) = ∑_{j} N_{λ_{j}^{−1}ϵ_{j}}(A_{j}) .

Write N_{ϵ} = ϵ^{−d} + f(ϵ). Because

\frac{log(N_{ϵ})}{log(ϵ^{−d})} = \frac{log(ϵ^{−d} + f(ϵ))}{log(ϵ^{−d})} → 1 ,

we must have f(ϵ)ϵ^{d} → 0. This leads to N_{λ_{j}^{(−1)}ϵ} = λ_{j}^{d}ϵ^{−d} + f(λ_{j}^{−1}ϵ_{j})). For ϵ → 0, we get

ϵ^{−d} + f(ϵ) = ∑_{j} λ_{j}^{d}ϵ^{−d} + f(λ_{j}^{−1}ϵ))

which gives after multiplication with ϵ^{d} and in the limit ϵ → 0 the claimed formula 1 = ∑ λ_{j}^{d}. □

One calls the unique solution of ∑_{j} λ_{j}^{d} = 1 the similarity dimension of the attractor of the iterated function system. It is in general not true that the similarity dimension and the Hausdorff dimension coincide but dim_{H}(A) ≤ dim_{S}(A) is always true Moran (in R) and Hutchinson (in R^{n}) have already shown the following

Theorem 3.6.2 (Moran 1946, Hutchinson 1981) Assume we have an iterated function system generated by similarities in the metric space R^{m}. Assume, there exists an open set U such that f_{i}(U)∩f_{j}(U) = ∅ for i ≠ j and U ⊃ f_{i}(U), then the similarity dimension of the attractor is the same as the Hausdorff dimension.

The proof is not so easy and can be found in the paper [32] of Hutchinson.

Remarks:

1) In the special case is when λ_{j} = λ is constant, one can solve 1 = n·λ^{−d} and the similarity dimension is given by

d = −\frac{log(n)}{log(λ)} .

In this case, the proof of the theorem of Moran and Hutchinson can be done along the lines of Exercice 3. The inequality dim_{H}(K) ≤ d can be shown by constructing explicit coverings. The other inequality is obtained by constructing a probability measure μ on K such that every point in K has an upper d− dimensional density which is bounded by a constant.

2) The box counting dimension and the Hausdorff dimension are not the same for all compact sets: The compact set A = {0} ∪ ⋃_{n}{1/n} ⊂ R has Hausdorff dimension 0 but box counting dimension dim_{B}(A) = 1/2.

5) The condition that f_{j}(U) are disjoint is called Morans open set condition. It is necessary in the theorem of Moran-Hutchinson as the following example shows: [0, 6] = [0, 4] ∪ [2, 6] = [0, 3]∪[3, 6] gives the two different similarity dimensions, namely the solutions of 2·(1/2)^{s} = 1 and 2 · (1/3)^{s} = 1.

3.7 Dynamical systems on fractals

Define on the space

A_{n} = {1, …, n}^{N} = {(α_{1}, α_{2}, …,) | α_{i} ∈ {1, … n}}

the metric d(α, β) = 2^{−N}, where N is the largest non-negative integer such that α_{i} = β_{i} for all i < N. The attractor K of an iterated function system f_{1}, f_{2}, … f_{n} can be labeled by elements in K_{n}. This labeling is sometimes unique.

Lemma 3.7.1 There exists a surjective continuous map ϕ : A_{n} ↦ K defined by

ϕ(α) = lim_{k→∞} ϕ_{k,α}(y) = lim_{k→∞} f_{α_{1}} ∘ f_{α_{2}} ∘ … f_{α_{k}}(y)

which is independent of y ∈ X. If the sets K_{j} = f_{j}(K) are disjoint, then ϕ is a homeomorphism of A_{n} onto K.

Proof. (i) By induction, one shows that for any pair y, z ∈ X and any α ∈ A_{n}

d(ϕ_{k,α}(y), ϕ_{k,α}(z)) ≤ λ^{k} · d(y, z) .

(ii) As in the proof of Banach’s fixed point theorem, we obtain for all y ∈ X

d(y, ϕ_{n,α}y) ≤ \frac{1}{1 − λ} · max_{j} d(y, f_{j}(y)) =: C .

(iii) For all y ∈ X, y_{n} = ϕ_{n,α}(y) is a Cauchy sequence in X and has a limit ỹ: indeed by (i),(ii),

d(y_{n}, y_{n+k}) ≤ λ^{n} · d(y, y_{k}) ≤ λ^{n} · C .

(iv) The limit ỹ does not depend on y because d(ϕ_{k,α}(y), ϕ_{k,α}(z)) ≤ λ^{k} · d(y, z) → 0. We call the limit ϕ(α).

(v) Because ϕ_{k,α}({y}) ⊂ F^{k}({y}) and F^{k}({y}) → K in the Hausdorff metric, it follows that ϕ(α) ∈ K. (vi) The map ϕ is surjective: given y ∈ K. By definition F^{k}({y}) → K. Any point z ∈ F^{k}({y} can be written as f_{α_{1}^{k}}f_{α_{2}^{k}} … f_{α_{k}^{k}}(y). Denote with α an accumulation point of

α^{(k)} = (α_{1}^{(k)}, α_{2}^{(k)}, … α_{k}^{(k)}, 1, 1, 1 …). Any such accumulation point gives ϕ(α) = y. (vii) We prove now that if K_{j} = f_{j}(K) are pairwise disjoint, the map ϕ is injective. Given α ≠ β. Denote with k the smallest index with α_{k} ≠ β′_{k}. Then ϕ(α) ∈ f_{α_{k}}(K) and ϕ(β) ∈ f_{β_{k}}(K) and ϕ(α), ϕ(β) are different. (viii) Because any bijective continuous map from the compact metric space A onto an other metric space B is a homeomorphism, the map ϕ is a homeomorphism if K_{j}, K_{i} are pairwise disjoint. □

The one-sided shift on A_{n} is defined as (T x)_{n} = x_{n+1}. It is a continuous map on A_{n} but not invertible.

Corollary 3.7.2 If f_{j}(K) are pairwise disjoint, there exists a continuous map S : K → K which is isomorphic to the one-sided shift T on A_{n}. Each of the maps f_{j} is a right inverse of S in the sense that S ∘ f_{j}(y) = y.

Proof. Transport the shift from A_{n} onto K by defining

S(y) = ϕ ∘ T ∘ ϕ^{−1}(y) .

Because ϕ^{−1} ∘ f_{j}(y) = (j, ϕ^{−1}(y)), we have

T ∘ ϕ^{−1}f_{j}(y) = ϕ^{−1}(y)

and so S(y) = ϕ ∘ T ∘ ϕ^{−1}f_{j}(y) = y. □

3.8 Skewproduct of contractions

We discuss now a generalisation of the concept of an iterated function system. The idea is to see the usual iterated function system as a ”skew product” system over the shift.

Given a topological dynamical system (X, T), where X is a compact metric space and T is a homeomorphism of X. Given an other metric space (Y, d). A continuous map A from X to the space of contractive maps on (Y, d) is called a contractive cocycle.

Given a contractive cocycle A over the dynamical system (X, T), we define a new dynamical system on X × Y by

F (x, v) → (T x, A(x)v) .

It is called a skew product. It is a continuous map on (X × Y). Denote by π : X × Y → Y, (x, v) ↦ v the projection onto the second coordinate.

Lemma 3.8.1 Let F be a skew product of a contracting cocycle. For all x ∈ X,

K(x) = lim_{n→∞} π(F^{n}(T^{−n}x, v)) ∈ Y

exists, is independent of v and defines a continuous map from X to Y.

Proof. Denote by V the complete metric space X := C(X, Y) of all continuous maps from X to Y. A metric δ on X is given by d(f, g) = sup_{x∈X} d(f(x), g(x)). The map F induces a continuous map F on X given by

F(v)(x) = πF (T^{−1}x, v(T^{−1}x)) = Av(T^{−1}x) .

This map F is a contraction because

||F(v) − F(w)||_{∞} ≤ λ · ||v − w|| .

It maps a ball of large radius into itself. Banach’s fixed point theorem assures that F has a unique fixed point. □

The image K = {F (x) | x ∈ X} of the fixed point is called the attractor of the contractive cocycle. It is compact, because it is the image of a continuous map from a compact metric space into a metric space.

The above construction generalizes iterated function systems. The dynamical system is the shift over n symbols. Given n contractions f_{i} : Y → Y. Define the cocycle A(x) = f_{x_{0}}. The attractor of the iterated function system f_{1}, f_{2}, … f_{n} is the same as the attractor of the cocycle A.

3.9 The Hutchinson operator on measures

Let (X, d) be a metric space. Denote with M the space of Borel probability measures on X. This space is a compact metric space with respect to the metric

d(μ, ν) = sup_{f∈Lip_{1}} ∫_{X} f(x)(dμ(x) − dν(x)) ,

where Lip_{1} = {f : X ↦ R | |f(x) − f(y)| ≤ d(x, y)}. A weighted iterated function system is given by n contractions f_{1}, …, f_{n} and n real positive numbers p_{j} satisfying ∑_{j=1}^{n} p_{j} = 1. The Hutchinson operator M : M → M is defined as

M(μ)(Y) = ∑_{j=1}^{n} p_{j}μ(f_{j}^{−1}(Y)) .

The support supp(μ) of a measure μ is defined to be the set of points x ∈ X such that for all ϵ > 0, the ball B(x, ϵ) has positive measure. The set supp(μ) is closed.

Lemma 3.9.1

a) M is a contraction on M and has a unique fixed point μ. b) The support of μ is the attractor K of the iterated function system f_{1}, …, f_{d}.

Proof. a) Denote by λ the minimum of the contraction factors of f_{j}.

d(M(μ), M(ν)) = sup_{f∈Lip_{1}} ∫ fd(M(μ) − M(ν))
= sup_{f∈Lip_{1}} ∫ ∑_{j} p_{j}f ∘ f_{j}d(μ − ν) .

For f ∈ Lip_{1} one has M^{∗}(f) = λ^{−1} · ∑_{j} p_{j}f ∘ f_{j} ∈ Lip_{1}. We can write

d(M(μ), M(ν)) = λ · sup_{f∈M^{∗}(Lip_{1})} ∫ fd(M(μ) − M(ν))
≤ λ · sup_{f∈Lip_{1}} ∫ fd(M(μ) − M(ν)) = λ · d(μ, ν) .

b) Denote with B the support of M(μ) = μ and with K the fixed point of F. Letting running the weighted iterated function system on the metric space (K, d) gives a fixed point ν which must coincide with μ. It follows that B ⊂ K. Given a point y ∈ K. We have to show that μ(B(y, ϵ)) > 0 for all ϵ > 0. Denote with x an element in ϕ^{−1}(y) ⊂ X_{n} labelling y. Because ϕ_{x,k}(y)(K) ⊂ B(y, ϵ) for sufficiently large k, one gets μ(ϕ_{x,k}(y)(K)) ≥ p_{x_{1}}p_{x_{2}} … p_{x_{k}} and so μ(B(y, ϵ)) > 0. □

Roughly speaking, the support of a measure is the set on which the measure ”lives”. If one thinks of the measure as a colored picture, then the support is the black and white analogue and it contains much less information than the measure.

Remark. the map supp : M → C is not continuous as one can already see on a metric space with two elements.

The fixed point μ = F (μ) is again called the attractor of the weighted iterated function system.

The following theorem which we will not prove here gives a possibility to construct the attractor μ of a weighted iterated function system f_{1}, f_{2}, … f_{n}, p_{1}, p_{2}, … p_{n}. Denote with ({1, …, n}^{N}, T, m) the one-sided Bernoulli shift, where the measure m is defined by m({x_{k} = i}) = p_{i}. Denote with f^{n}(x) the map f_{x_{n}} ∘ … ∘ f_{x_{1}} and with f^{0}(x) the identity.

Theorem 3.9.2 (Elton’s ergodic theorem) Denote by μ the attractor of M. For all continuous functions g ∈ C(X) and almost all x ∈ {1, …, n}^{N} one has

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} g(f^{n}(x))(y) = ∫ g dμ .

3.10 Overview over different dimensions

Topological dimension: (or small inductive dimension)

Defined for any topological spaces (X, U). Takes values in {−1, 0, 1, 2, …, ∞}. Definition: dim_{T} (X) = 0 ⇔ ∃ basis V ⊂ U and ∀V ∈ V is clopen. Define inductively dim_{T} (X) ≤ k ⇔ exists basis V ⊂ U such that ∀V ∈ Vdim_{T} (δV) ≤ (k − 1). dim_{T} (X) = k ⇔ dim_{T} (X) ≰(k − 1) and dim_{T} (X) ≤ k. Property: is invariant under homeomorphisms.

Hausdorff dimension:

Defined for all subset of a metric space (X, d). Takes values in [0, ∞]. Definition:

h^{s}(A) = lim_{ϵ→0} \begin{matrix}inf \\ U & ϵ \end{matrix} ∑_{U∈U_{ϵ}} |U|^{s} .

dim_{H}(A) is the value s, where h^{s}(A) jumps from ∞ to 0. Property: is invariant under metric isomorphisms.

Box counting dimension: (or entropy dimension or fractal dimension)

Defined for some compact subsets of a metric space (X, d). Takes values in [0, ∞]. Definition: N_{ϵ}(A) smallest number of sets with diameter ϵ covering A.

dim_{B}(A) = lim_{ϵ→0} \frac{log(N_{ϵ}(A))}{− log(ϵ)} .

Property: is invariant under metric isomorphisms.

Similarity dimension:

Defined for attractors A of iterated function systems f_{j} with contraction rates λ_{j}. Takes values in (0, ∞). Definition: dim_{S}(A) = solution d of ∑_{j} λ_{j}^{d} = 1. Property: is invariant under isometries.

3.11 Epilog

Why studiing fractals?

• Fractals are fascinating objects.

• Many objects in nature are fractal in nature at least in some range of scales.

• There are known simple dynamical systems like the quadratic map z ↦ z^{2} +E which have invariant sets which are fractal.

• Some subsets of parameter planes like the Mandelbrot set look fractal. The Mandelbrot set M satisfies however dim_{H}(M) = 2.

• Fractals can be considered as fixed points of maps defined on the space of compact sets of the plane.

• There is evidence that complicated dynamical systems like partial differential equations describing the weather have a dynamics which can be described as a flow on a finite dimensional strange attractor which is a fractal.

• Selfsimiliary is an intersting kind of symmetry. Objects which are invariant under some symmetry operations have always been interesting in mathematics. Fractal geometry is therefore a natural extension of classical geometry in the sense of Klein’s erlanger programm.

• From the topological point of view, the Cantor set E, the prototype of a fractal is important: every nonempty compact perfect Hausdorff space with a countable basis consisting of clopen sets is homeomorphic to the Cantor set. Furthermore, every compact metric space is the image of a continuous map from the Cantor set E.

A bit of history: The word fractal was coined by Benoit Mandelbrot in the 1970’ies. (The word ’fractal’ was derived from the latin fractus meaning broken.) Historical examples of fractal objects were developed by mathematicians in the early years of the century. The Cantor set was not invented by Cantor (1845-1918) but by Henry Smith in 1875. Cantor used this invention in 1883. (This reference can be found in Ian Stewart’s book [57] where he asks ”Let’s face it, Smith’s set isn’t very impressive, is it?) Fractals objects appeared in paintings of Albrecht Dürer (1471-1528) and Maurits Escher (1902-1972) (see http://server.berkeley.edu/Escher/). Mathematicians who studied fractal objects were Helge von Koch (1870-1924), Waclaw Sierpinski (1882), Gaston Julia (1893-1978) and Pierre Fatou (1878-1929). The idea of the iterated function system is very natural and also quite old: It goes back to a paper of P. Moran in 1946 ([32]) who studiied iterated function systems on R. Barnsley’s book [3] is a successful attempt to popularize Hutchinson’s paper. and contains also additional new ideas. We recommend also the books [19, 22, 52] for the mathematics of fractals.

3.12 Exercices

Topic: The Hausdorff dimension of the Cantor set

1) Define E_{0} = [0, 1], E_{1} = [0, 1/3] ∪ [2/3, 1], E_{2} = [0, 1/9] ∪ [2/9, 3/9] ∪ [6/9, 7/9] ∪ [8/9, 1] and inductively E_{j+1} by removing the middle third in each interval of E_{j} so that E_{j} = ⋃_{l=1}^{2^{j}} E_{j}^{(l)} consists of 2^{j} intervals E_{j}^{(l)} of length 3^{−j}. The set E = ⋂_{j=0}^{∞} E_{j} is called the Cantor set. Show that it is a compact subset of R. Show that E is homeomorphic to {0, 1}^{N}.

2) Define ϵ_{k} = 3^{−k}. Show that

h_{ϵ_{k}}^{s}(E) ≤ 2^{k} · (ϵ_{k})^{s} .

3) Show that for s = log(2)/ log(3)

h^{s}(E) ≤ 1 .

and conclude that dim_{H}(E) ≤ s.

4) There exists a measure μ on E such that for all E_{j}^{(l)} ⊂ E_{j}

μ(E_{j}^{(l)}) = 2^{−j} .

You can assume the existence of the product measure on {0, 1}^{N} and use (1).

5) Use a result in the course to show that dim_{H}(E) ≥ s. You have proven:

dim_{H}(E) = \frac{log(2)}{log(3)}.

Billiards

4.1 The billiard map

A curve Γ in R^{2} is a continuous map γ : R → R^{2}. The curve is called smooth, if the map γ : R → R^{2} is arbitrarily often differentiable. If the curve is closed, γ(t + T) = γ(t) for all t ∈ R and some T > 0. A closed smooth curve can always be parametrized by a new time q such that the length of the piece between γ(q) and γ(q′) is q − q′ and γ(q + 2π) = γ(q). The curve γ is then a continuous map from the circle R/(2πZ) into R^{2}. We write in future for simplicity just q instead of γ(q). The tangent argument ϕ(q) is the angle between the tangent line at the point q and the first coordinate axis. (See Fig.1) The real number κ(q) = \frac{d}{dq} ϕ(q) called the curvature at a point q ∈ T^{1}. (See Fig. 7.) A smooth closed curve Γ is called convex, if κ(q) ≥ 0 for all q.

Given a smooth convex billard table which is a smooth convex closed curve Γ = {γ(q) | q ∈ T^{1} = R/(2πZ)}. Given two different points q, q′ ∈ Γ, we denote with L(q, q′) the line passing through q and q′. (See Fig 2.) In the limiting case q = q′, L(q, q) denotes the tangent line at the point q. Denote with ϕ(q, q′) the angle between the tangent L(q, q) and L(q, q′). Let ϕ′(q, q′) be the angle between the line L(q, q′) and the tangent L(q′, q′). The convexity of the curve implies that for all pairs q ≠ q′, there exists a unique point q′′ ∈ Γ such that ϕ′(q, q′) = ϕ(q′, q′′). For q = q′, define q′′ = q. There exists therefore a map

T̃ : (q, q′) → (q′, q′′)

on T^{2}. We will rewrite now this map in other coordinates: Define

p(q, q′) = cos(ϕ(q, q′)), p′(q, q′) = cos(ϕ′(q, q′)) .

Given (q, p), we can calculate q′ such that p(q, q′) = p and the above map T̃ can be rewritten as

T : (q, p) ↦ (q′, p′) .

This is called the billiard map. In these coordinates, it is a map on the annulus T^{1} ×[−1, 1].

The billiard map can also be defined in the case when Γ is only piecewise smooth or when the billiard table is no more assumed to be convex. In such cases, there are countably many points (q, p), where some T^{n}(q, p) is not defined.

Numerical calculation of the billiard map with the computer

<<"Graphics`ImplicitPlot`"
Size[{a_,b_}]:=a^2 + b^2; Normed[ww_]:=ww/Sqrt[Size[ww]]
Curve=x^4+y^4-1; Inside[{a_,b_}]:=(Curve /. {x->a, y->b}) < 0;
Grad=Normed[{D[Curve,y],-D[Curve,x]}];
T[{P_,v_}]:=Module[{H=P,L=P,m}, While[Inside[H],H+=0.01 v];
  While[Size[H-L]>10^(-4),M=(H+L)/2;If[Inside[M],L=M, H=M]];
  w=Grad /. {x->L[[1]],y->L[[2]]}; {L,Normed[2 (v.w) w-v]}];
Orb=NestList[T,{{0,.2},{.707,.707}},30];
Show[Graphics[Line[Table[Orb[[i,1]],{i,Length[Orb]}]]],
     ImplicitPlot[Curve==0,{x,-1,1}],PlotRange->{{-1,1},{-1,1}}];

See Fig. 9. For the output of the program.

4.2 The generating function

The billiard map was defined by requiring

ϕ′(q, q′) = ϕ(q′, q′′) .

This is the natural reflection law ”angle of incidence=angle of reflection”. We define now a smooth function h(q, q′) as the oriented length of the straight line-segment between q and q′. We calculate

h_{2}(q, q′) = − cos(ϕ′(q, q′)) = −p′(q, q′), h_{1}(q, q′) = cos(ϕ(q, q′)) = p(q, q′)

(see Fig. 3) so that the equation p(q′, q′′) = p′(q, q′) defining the billiard map can be written as

h_{1}(q′, q′′) + h_{2}(q, q′) = 0 .

The function h(q, q′) is called the generating function of the billiard map.

Lemma 4.2.1 The map T : (q, p) ↦ (q′, p′) is area-preserving: for any measurable Y ⊂ X = T^{1} × [−1, 1], one has

∫_{Y} dp dq = ∫_{Y} dp′ dq′

Proof. It suffices to prove the claim for open balls Y because they generate the Borel σ algebra. Using Green’s formula, we get

∫ ∫_{T}^{−1}(Y)dp dq = ∫_{T^{−1}(δY)} p dq
= ∫_{T^{−1}(δY)} h_{1}(q, q′) dq = ∫_{δY} h_{1}(q′, q′′) dq′
= ∫_{δY} −h_{2}(q, q′) dq′ = ∫_{δY} p′ dq′
= ∫ ∫_{Y} dp′ dq′ .
□

Having preserved a finite measure m = dqdp on T^{1} × [−1, 1], we can apply the following recurrence theorem of Poincare:

Theorem 4.2.2 (Poincaré recurrence theorem) Given a measure-preserving invertible transformation T on a probability space (X, m). For any measurable set Y of positive measure, there exists n ∈ Z such that m(Y ∩ T^{n}(Y)) > 0.

Proof. We argue indirectly. Suppose the claim is wrong. There exists then a set Y of positive measure such that Y_{i} = T^{i}(Y) satisfies m(Y_{i} ∩ Y) = 0 for all i > 0. Because m(Y_{i}) = m(Y) > 0, there must exist 0 < i < j such that m(Y_{i} ∩ Y_{j}) > 0. But then also m(T^{−i}(Y_{i} ∩ Y_{j})) = m(Y ∩ Y_{j−i}) > 0. Contradiction. □

Remark. Generating functions exist in general for area preserving twist maps. An example is the conservative Hénon map

T : ( \begin{matrix}q \\ p \end{matrix} ) ↦ ( \begin{matrix}q^{2} − c − p \\ q \end{matrix} ) .

An orbit (q_{n}, p_{n}) = (q_{n}, q_{n−1}) satisfies the equation

q_{n+1} − 2q_{n} + q_{n−1} = q_{n}^{2} − c − 2q_{n} =: V′(q_{n})

This can be written as

h_{1}(q_{n}, q_{n+1}) + h_{2}(q_{n−1}, q_{n}) = 0

with h(q, q′) = −\frac{(q′−q)^{2}}{2} − V (q).

4.3 Existence of periodic orbits

We write X = T^{1} × [−1, 1] and x = (q, p). If there exists n ∈ N such that T^{n}(x) = x, then x is called a periodic point of the map T : X → X.

Looking for periodic orbits of billiards is equivalent to find a critical point ∇L(q) = 0 of a function L : R^{n} → R:

Proposition 4.3.1 (q_{1}, p_{1}), (q_{2}, p_{2}), …, (q_{n}, p_{n}) = (q_{1}, p_{1}) is a periodic orbit of the billiard map, if and only if (q_{1}, …, q_{n}) is a critical point of the functional

L(q_{1}, …, q_{n}) = ∑_{i=1}^{n} h(q_{i}, q_{i+1}) .

Proof. The gradient of the function L is

∇_{k}L(q_{1}, …, q_{n}) = h_{1}(q_{k}, q_{k+1}) + h_{2}(q_{k−1}, q_{k}) ,

where the indices k are taken modulo n so that q_{n+1} = q_{1}, q_{0} = q_{n}. □

This proposition gives a tool to construct periodic orbits of period n by maximizing the length of a n-polygon inscribed in the curve Γ.

Corollary 4.3.2 (Birkhoff) Assume γ is differentiable and convex. For any n ≥ 2, there exists a periodic orbit of period n.

Proof. The continuous function L takes its maximum on the compact set T^{n}. Each point (q_{1}, q_{2}, …, q_{n}) ∈ T^{n} represents a polygon and L is the length of the polygon given by the billiard ball track. The point (q_{1}, q_{2}, … q_{n}) where L is maximal is a critical point of L. It is therefore an orbit of the billiard map. The maximal polygon must have at least n sides because we could else increase the length by replacing a line q_{n}q_{n+1} by two lines q_{n}qq_{n+1} with q on the arc between q_{n} and q_{n+1}. This would increase the length of the polygon. (see Fig. 4.) □

Remark. One could also try to find periodic orbits by minimizing the length of the polygon. This is however not good, because the minimal polygon has no more n vertices for n > 2.

4.4 Classification of periodic orbits

We denote by dT the Jacobian of a map T. Jacobian The map T is called orientation preserving, if det(T (x)) > 0 for all x ∈ X.

Lemma 4.4.1 det(dT (x)) = 1 if and only if T is orientation-preserving and area preserving.

Proof. We have for any subset Y of R^{2} the transformation rule

∫_{T^{−1}(Y)} dx = ∫_{Y} d(T x) = ∫_{Y} |det(T x)|dx .

If det(dT (x)) = 1, then T is by definition orientation-preserving and by the transformation rule T is area preserving. If T is area preserving and orientation-preserving, then |det(x)| = 1 and det(x) > 0 and so det(x) = 1. □

Corollary 4.4.2 For any n, dT^{n} has the two eigenvalues λ, λ^{−1}.

Proof. The eigenvalues λ_{±} of dT^{n} satisfy λ_{−} · λ_{+} = det(dT^{n}) = 1. □

A periodic point x is called elliptic, if |λ| = 1, λ ≠ 1, parabolic, if λ = ±1 and hyperbolic, if |λ| ≠ ±1.

Lemma 4.4.3 A periodic point x = T^{n}(x) is elliptic if |trace(dT^{n})(x)| < 2, it is parabolic, if trace(d^{n}(x)| = 2 and hyperbolic, if |trace(dT^{n})(x)| > 2.

Proof. The eigenvalues λ_{±} of dT^{n} are the solutions of λ^{2} + trace(dT^{n})λ + 1 = 0. These solutions are

trace(dT^{n})(x)/2 ± \sqrt{(trace(dT^{n})(x)/2)^{2} − 1} .
□

The Hessian at a critical point is the matrix

L_{ij} = \frac{d}{dq_{i}} \frac{d}{dq_{j}} L(q_{1}, …, q_{n}) .

One calculates

L = ( \begin{matrix}b_{1} & a_{1} & 0 & · & · & a_{n} \\ a_{1} & b_{2} & a_{2} & · & · & · \\ 0 & a_{2} & · & · & · & · \\ · & · & · & · & a_{n−2} & 0 \\ · & · & · & a_{n−2} & b_{n−1} & a_{n−1} \\ a_{n} & · & · & 0 & a_{n−1} & b_{n} \end{matrix} ) ,

where a_{i} = h_{12}(q_{i}, q_{i+1}) and b_{i} = h_{11}(q_{i}, q_{i+1}) + h_{22}(q_{i−1}, q_{i}). This matrix L is called a periodic Jacobi matrix. For the billiard map, one gets

a_{i} = \frac{sin(ϕ_{i}) sin(ϕ_{i+1})}{l_{i}}, b_{i} = sin^{2}(ϕ_{i})(\frac{1}{l_{i}} + \frac{1}{l_{i−1}} ) − 2 sin(ϕ_{i})κ_{i} ,

where l_{i} = h(q_{i}, q_{i+1}) are the lenghts of the secants, ϕ_{i} = ϕ(q_{i}, q_{i+1}) and κ_{i} = κ(q_{i}) are the curvatures at the reflection points. (See Fig 6. and 7. for the derivation of these formulas.)

We want now to find the Jacobean of the map T. It turns out that dT is quite ugly and that it is better to choose other coordinates in which dT can be understood also geometrically. Start with the map

T̃ : x_{i} = (q_{i}, q_{i−1}) → x_{i+1} = (q_{i+1}, q_{i})

determined by h_{1}(q_{i}, q_{i+1} + h_{2}(q_{i−1}, q_{i}) = 0. Differentiate this equation with respect to q_{i}, q_{i−1} gives dq_{i+1}/dq_{i} = −b_{i}/a_{i}, dq_{i+1}/dq_{i−1} = −a_{i−1}/a_{i}. The Jacobean of T̃ is therefore

dT̃(x_{i}) := ( \begin{matrix}−b_{i}/a_{i} & −a_{i−1}/a_{i} \\ 1 & 0 \end{matrix} ) .

(It is also called the transfer matrix for the operator L). We see that the Jacobean of the map T̃ can be given without knowing explicitly the formula for the map T̃.

The matrix dT̃(x_{i}) requires the knowledge of two lengths, a curvature and two angles. There exist basis vectors (e_{1}(x), e_{2}(x)) in which the Jacobean is nicer.

Proposition 4.4.4 There exist coordinates such that the Jacobean of the billiard map T : (q_{i−1}, p_{i−1}) ↦ (q_{i}, p_{i}) is given by the matrix

B_{i} = ( \begin{matrix}1 & 0 \\ −\frac{2κ_{i}}{sin(ϕ_{i})} & 1 \end{matrix} ) · ( \begin{matrix}1 & l_{i} \\ 0 & 1 \end{matrix} ) .

Proof. With a first coordinate transformation F_{i} = ( \begin{matrix}a_{i}^{−1} & 0 \\ 0 & 1 \end{matrix} ) we can acchieve that the

determinant is 1:

F_{i}^{−1}dT̃F_{i−1} = A_{i} = (a_{i−1})^{−1} ( \begin{matrix}−b_{i} & −a_{i−1}^{2} \\ 1 & 0 \end{matrix} )

One calculates then with G_{i} = ( \begin{matrix}0 & − sin(ϕ_{i}) \\ 1/ sin(ϕ_{i}) & sin(ϕ)/l_{i} \end{matrix} )

G_{i}^{−1} · A_{i} · G_{i−1} = ( \begin{matrix}1 & l_{i} \\ −\frac{2κ_{i}}{sin(ϕ_{i})} & 1 − \frac{2κ_{i}}{sin(ϕ_{i})} \end{matrix} ) = ( \begin{matrix}1 & 0 \\ −\frac{2κ_{i}}{sin(ϕ_{i})} & 1 \end{matrix} ) · ( \begin{matrix}1 & l_{i} \\ 0 & 1 \end{matrix} )

which is the product of two matrices, one being the Jacobean of the translation involving the length of the ball until it hits the boundary and the second the Jacobean of the reflection involving the curvature at the impact point and the angle of impact. □

This formula for the linearisation of the billiard map is also true for billiards in not necessarily convex regions, where there is in general no more a generating function.

The value g_{i} = \frac{sin(ϕ_{i})}{2κ_{i}} has a geometrical interpretation. (See Fig. 8.) Draw the normal at the point q and draw a circle with radius \frac{1}{2κ_{i}} lying on the normal and being tangent to Γ. Then g_{i} is the length of the billiard ball track inside the interior of the circle.

4.5 The formula of Green,Mackay,Meiss

The stability of a periodic orbit can be read off from the determinant of the matrix L and the product ∏_{i=1}^{n} a_{i}. This is the consequence of the following relation which is called the formula of Green,MacKay,eiss,

Lemma 4.5.1

trace(dT̃^{n}) − 2 = \frac{det(L)}{∏_{i=1}^{n} a_{i}}

Proof. Call λ, λ^{−1} the eigenvalues of the dT̃^{n}.

det(dT̃^{n} − ξ)

is a polynomial in ξ + ξ^{−1} which vanishes for ξ = λ, λ^{−1}. This implies that

det(dT̃^{n} − ξ) = (ξ + ξ^{−1}) − tr(dT̃^{n}) .

The zeros of

det(L(ξ)) = det ( \begin{matrix}b_{1} & a_{1} & 0 & · & \frac{a_{n}}{ξ} \\ a_{1} & b_{2} & 0 & · & 0 \\ · & · & · & · & · \\ 0 & 0 & · & b_{n−1} & a_{n−1} \\ a_{n}ξ & 0 & · & a_{n−1} & b_{n} \end{matrix} ) ,

are the eigenvalues of dT̃^{n}. So

det(L(ξ)) = C · ((ξ + ξ^{−1}) − tr(dT̃^{n})) ,

where C is a constant. Because we have for ξ → ∞

det(L(ξ)) = −(∏_{i=1}^{n} a_{i})ξ + ...

the constant is C = − ∏_{i=1}^{n} a_{i} and

det(L(ξ)) = −(∏_{i=1}^{n} a_{i}) · (ξ + ξ^{−1} − tr(dT̃^{n})) .

The claim follows with ξ = 1. □

We can say that the stability problem of a periodic orbit is a spectral problem for a periodic Schrödinger operator.

4.6 Caustics

A curve γ in the interior of the billiard table is called a caustic, if a billiard trajectory which is once tangent to this curve stays tangent. Formally, this means that if a line L(q, q′) is a tangent to γ and if T̃(q, q′) = (q′, q′′), then L(q′, q′′) is tangent of γ. It follows that to each q ∈ T^{1} there exists an angle ϕ_{γ}(q) such that the ball starting with this angle at the point q of Γ is tangent to the caustic.

Lemma 4.6.1 The invariant curve {(q, ϕ_{γ}(q))} ⊂ X corresponding to the caustic divides the annulus X into two connected regions X_{1}, X_{−1} both being invariant under the billiard map.

Proof. The three curves γ_{−1} = {(q, −1) | q ∈ T^{1}}, γ_{1} = {(q, 1) | q ∈ T^{1}} and γ_{0} = {(q, ϕ_{γ}(q)) | q ∈ T^{1}} are invariant under the billiard map. Call X_{−1} the region beween γ_{−1} and γ_{0} and X_{1} the region between γ_{1} and γ_{0}. Assume, there exists a point (q, p) ∈ X_{−1} which is mapped into X_{1}. Connect (q, p) with (q, −1) by a straight line. By the continuity of the map and the intermediate value theorem, there exists a point (q, t) on this interval, which is mapped onto γ_{0}. By the invariance of γ_{0} by the billiard map T and its inverse T^{−1}, the point (q, t) must also ly on γ_{0}. □

The billiard map is called ergodic, if every T invariant measurable set of X has either measure 1 or 0.

Corollary 4.6.2 The existence of a caustic prevents the billiard map from being ergodic.

If an annular region invariant region contains no caustics, this region is called a domain of instability.

Proposition 4.6.3 (Birkhoff) Given a caustic γ of a billiard map at a smooth convex table Γ. Then the curve ϕ_{γ} ⊂ X is Lipschitz continuous. On this curve, S(q_{n}) = q_{n+1} and S^{−1}(q_{n}) = q_{n−1} are almost everywhere differentiable functions.

We don’t prove this result here. A proof can be found in [?].

4.7 Lyapunov exponents

Given a measurable 2 × 2 matrix-valued function B and a dynamical system (X, T, m). Define

B^{k}(x) := B(T^{k−1}x)B(T^{k−2}x) … B(x) .

If the limit

λ(x) = lim_{n→∞} ∫_{X} \frac{1}{n} log(||B^{n}(x)||) dm(x) .

exists, it is called the Lyapunov exponent of B. For a periodic orbit x with period n, this limit clearly exists and coincides with the eigenvalue of B^{n}(x). In general, it exists only almost everywhere:

Proposition 4.7.1 (Fürstenberg-Kesten) Let (X, T, m) be a measure preserving dynamical system and let A be a measurable matrix valued function satisfying ∫_{X} log ||B(x)|| dm(x) < ∞. Then, λ(x) exists for almost all x ∈ X and is finite.

It is in general a difficult problem to prove positive Lyapunov exponents.

Lemma 4.7.2 (Wojtkowski) If A is a measurable 2 × 2-matrix-valued function with det(A(x)) = 1 and [A]_{ij}(x) ≥ ϵ, then

λ(A) ≥ \frac{1}{2} log(1 + 2ϵ^{2}) .

Proof. Define the function F : R^{2} → C by

v = (v_{1}, v_{2}) ↦ F (v) = (v_{1} · v_{2})^{1/2} .

For a matrix B with determinant 1 satisfying [B]_{ij}(x) ≥ ϵ, define

ρ(B) = inf_{F (v)=1} F (Bv) .

(i) Given a 2 × 2-matrix A satisfying [A]_{ij} ≥ ϵ. Then

ρ(A) ≥ (1 + 2ϵ^{2})^{1/2}

Proof. If A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and w = (w_{1}, w_{2}) with F (w) = (w_{1}w_{2})^{1/2} = 1 then

F (Aw) = (aw_{1} + bw_{2})^{1/2}(cw_{1} + dw_{2})^{1/2}
≥ (ad − bc + 2bc)^{1/2} ≥ (1 + 2ϵ^{2})^{1/2} .

(ii) ||B|| ≥ ρ(B) Proof. Take v = (1, 1). Then ||A|| ≥ \frac{|Av|}{|v|} ≥ \frac{F (Av)}{F (v)} ≥ ρ(A).

(iii) ρ(AB) ≥ ρ(A) · ρ(B). Proof.

ρ(AB) = inf_{F (v)=1} F (ABv) ≥ inf_{F (Bv)=1} \frac{F (ABv)}{F (Bv)} · inf_{F (v)=1} F (Bv) = ρ(A) · ρ(B) .

(iv) Proof of the claim: We get from (ii),(iii),(i) that

\frac{1}{n} log ||A^{n}(x)|| ≥ \frac{1}{n} log(ρ(A(T^{n−1}x) … ρ(A(x))) ≥ \frac{1}{n} log((1 + 2ϵ^{2})^{n/2}) .

□

Without proof we mention the generalisation of this result which says

Theorem 4.7.3 (Wojtkowski) If A is a measurable 2 × 2-matrix-valued function with det(A(x)) = 1 and [A]_{ij}(x) ≥ 0 and [A]_{ij}(x) > 0 on a set of positive measure, then

∫_{X} λ(A(x)) dm(x) > 0 .

This result is usually applied as follows for a differentiable map T : X → X: assume we can find for each point x ∈ X a basis (e_{1}(x), e_{2}(x)) such that the linear map dT (x) is in this base given by a matrix A(x) satisfying the condition of the theorem of Wojtkowski.

4.8 Positive Lyapunov exponents for billiards

We have found basis vectors (e_{1}(x), e_{2}(x)) such that the Jacobean dT (x) of the billiard map is given by the matrix

B(x) = B_{2}(x)B_{1}(x) = ( \begin{matrix}1 & 0 \\ −1/g(T x) & 1 \end{matrix} ) · ( \begin{matrix}1 & l(x) \\ 0 & 1 \end{matrix} ) ,

where g(x) = \frac{sin(ϕ(x))}{2·κ(x)} and l(x) is the length between q(x) and q(T x). The matrix B_{1}(x) is related with motion of the ball insided the table from q to q′ and involves the length l = h(q, q′) of that track. The matrix B_{2}(x) comes from the reflection at the boundary at q′ and involves the curvature κ(T x) and the angle ϕ(T x) of the impact corresponding to the point T (x) = (q′, p′).

We first have to assure the condition ∫_{X} log ||B(x)|| dm(x) < ∞ to be sure that the Lyapunov exponent exists by Fürstenberg-Kesten for almost all x ∈ X.

Lemma 4.8.1 Assume κ(x) is bounded by a constant then ∫_{X} log ||B(q, p)|| dpdq < ∞.

Proof. Because p = cos(ϕ) we have dp = − sin(ϕ)dϕ so that

∫ |1/g(q, p)| dpdq ≤ ∫ 2 · |κ(q)| dϕdq < ∞ .
□

We get directly from Wojtkowski’s lemma, that billiards with concave pieces of the table have positive Lyapunov exponents:

Corollary 4.8.2 Assume, there exists ϵ > 0 such that −κ(x) ≥ ϵ > 0 and l(x) ≥ ϵ > 0, then the Lyapunov exponent satisfies λ(x) ≥ \frac{1}{2} · log(1 + 2 · ϵ^{2}).

Proof. The matrix

( \begin{matrix}1 & l(x) \\ −\frac{2κ(T x)}{sin(ϕ(T x))} & 1 − \frac{2κ(T x)}{sin(ϕ(T x))} \end{matrix} ) =

has entries B_{ij}(x) ≥ ϵ. □

There are also convex billiards with positive Lyapunov exponents: unfortunately, no smooth convex billiards are known up today which have this property. One considers therefore piecewise smooth billiards.

Theorem 4.8.3 (Wojtkowsky) Assume, a piecewise smooth convex curve γ which has the property that for any two points q, q′ on the strictly convex parts of the curve

2g(q) + 2g(q′) ≤ l(q, q′)

with inequality on a set of positive measure, then the billiard map T at γ has positive Lyapunov exponents on a set of positive measure.

Proof. A vector v = (1, f) is maped by the matrix B_{1}(x) to the vector (1, f + l(x)). This vector is then maped by B_{2}(x) to the vector (1 − (f + l(x))/2g(T x), f + l(x)) which is after a rescaling of length the vector (1, \frac{(f+l(x))wg(T x)}{2g(T x)−f−l(x)}. If we don’t care about the length of the vector, the map v ↦ B(x)v is determined by the map

\overline{B} : f ↦ f + l ↦ \frac{1}{1/(f + l) − 1/g(T)} = \frac{(f + l)2g(T)}{2g(T) − f − l} .

At each point x ∈ X, we define a basis given by e_{2}(x) = (1, 0) and e_{1}(x) = (1, −g(x)). Claim: Assume 2g(x) + 2g(T x) ≤ l(x) with inequality on a set of positive measure. In this basis, the matrix B(x) is positive and there exists a set of positive measure, where [B(x)]_{ij} ≥ ϵ > 0 for some ϵ > 0. Proof. We have to show that the map \overline{B} maps the interval [0, −2g(x)] into the interval [0, −2g(T x)] and into its interiour for a set of positive measure. This is just a calculation:

\overline{B} : 0 ↦ \frac{l(x)2g(T x)}{2g(T x) − l(x)} ≥ −2g(T x) .

\overline{B} : −2g(x) ↦ \frac{(−2g(x) + l(x))2g(T x)}{2g(T x) + 2g(x) − l(x)} ≤ 0 .

□

The condition 2g(q, q′) + 2g(q′, q′′) < l(q, q′) is a geometric condition which can be checked geometrically. An example of a billiard satisfying this condition is the stadion. An other example is the a rectangle, with rounded corners.

4.9 Exterior billiards

Given a closed oriented convex curve Γ in the plane. Take a point P outside the curve, draw the positive tangent t at Γ and define the point T (P) to be the unique point on t different from P such that distance (P, Γ ∩ t) = distance (T (P), Γ ∩ t). We are interested in the map T : P ↦ T (P).

It is unknown if there exists convex curve Γ and a point P such that the orbit P, T (P), T ∘ T (P), T ∘ T ∘ T (P), … is unbounded. Here is a Mathematica simulation as shown in the announcement of the course.

Remark: For smooth curves Γ, it is known that an orbit cannot escape to ∞) because there are invariant curves arbitrarily far away from the given curve Γ.

We assume now Γ is smooth and convex. Given two angles q, q′. Denote with L(q) rsp. L(q′) the tangent at the curve Γ with makes the angle q rsp. q′ with the first coordinate axes. In the case, when L(q) and L(q′) intersect in a point P (q, q′), define h(q, q′) to be the area of the bounded region enclosed by Γ and the tangents having angle q, q′ Denote with r the distance between L(q) ∩ Γ and P (q, q′) and with r′ the distance between L(q′) ∩ Γ and P (q, q′). One checks that

p(q, q′) := −h_{1}(q, q′) = \frac{r^{2}}{2} , p′(q, q′) := h_{2}(q, q′) = \frac{r^{′2}}{2} .

The convexity of the curve implies that the tangent L(q′) is turning monotonically. This implies that h_{12}(q, q′) > 0. Applying the implicit function theorem assures that q′ and so also p′ can be written as a function of q, p. The map T : (q, p) ↦ (q′, p′) is called the exterior billiard map or Moser map Lemma 4.9.1 T is area-preserving.

Proof. The existence of the generating function assures that T is area-preserving. The proof is the same as in the billiard case. □

Remark. the invariant measure dpdq = rdrdϕ is the Lebesgue measure on the plane.

The function h is again called the generating function of the exterior billiard map. Periodic orbits of the exterior billiard can in the same way be found as critical points of the functional

L(q_{1}, …, q_{n}) = ∑_{i=1}^{n} h(q_{i}, q_{i+1}) :

Theorem 4.9.2

For any n ≥ 2, there exists a periodic orbit of period n.

L(q_{1}, …, q_{n}) = ∑_{i=1}^{n} h(q_{i}, q_{i+1}) .

Proof. Th function L is the difference between the area of the polygon spanned by the points P_{i} = P (q_{i}, p_{i}) and the fixed area of the region bounded by Γ. Consider all n-gons which have all sides tangent to Γ. Take the polygon with minimal area. Such a minimum cannot be a polygon with less edges because we can else decrease the area by adding an other tangent. □

Remarks.

1) We see that there exist many parallels between exterior and interior billiards. There are two main difference between the two systems.

a) The exterior billiard is defined on an not compact space.

b) the exterior billiard is affine invariant. Every affine equivalent curve to a given curve gives an isomorphic exterior billiard map. Because of a) there exists a priory no invariant probability measure which is absolutely continuous to the Lebesgue measure.

2) If one looks at billiards and exterior billiards on the sphere, where we take geodesics instead of straight lines, the two problems, exterior billiards and interior billiards become dual to each other.

4.10 Open research problems

• Two billiard tables have isomorphic billiard maps where the isomorphism is given by a differentiable map. Are the tables congruent? (Guillemin)

• If two billiard tables have the same spectrum for the Dirichlet Laplacian, then the maps are isomorphic.

• The set of periodic orbits has measure zero.

• Are periodic orbits dense? (An old question by Poincaré posed in the context of the three body problem.)

• Is there a smooth or even real analytic convex billiard which has positive metric entropy.

• The Birkhoff conjecture: is an integrable convex billiard map an ellipse.

4.11 Epilog

Generalisations of billiards.

• Billiards in higher dimensions.

• Billiards in a nonflat table, or in a gravitational field.

Why studiing billiards?

• Billiards form fascinating dynamical systems. which are easy to describe.

• One can think of billiards as a limit of geodesic flows.

• The motion of two hard discs on the torus factors into a Billiard flow and a quasiperiodic motion.

• There are connections between billiards and spectral problems of Laplacians with some boundary conditions on the table.

• They are prototypes for monotone twist maps occuring for example in celestial mechanics. Many things which can be proven for billiards can be proven also for such twist maps. Billiards are a good model problem.

• Billiards show intersting behaviour variing from integrable to chaotic motion.

• Various subjects of mathematics are involved while studiing billiards: ergodic theory, calculus of variations, Morse theory, KAM theory, spectral theory of discrete and continuous Schroedinger operators.

A little bit history:

We mention some mark stones in the development of billiards.

• 1927 G.D. Birkhoff invents the billiard system [7].

• 1970 Ya. G. Sinai studies dispersing billiards.

• 1973 V.F. Lazutkin proves the existence of caustics.

• 1979 L.A. Bunimovich invents the stadion.

• 1982 J. Mather proves the existence of smooth glancing billiards [42].

• 1986 M. Wojtkowski designs a large class of billiards with positive metric entropy [59].

• 1991 V. Donnay gives the existence of billiards with positive metric entropy for billiards having C^{1} smooth boundaries [18].

The exterior billiard appeared later in the literature.

• 1963 The problem is posed by P.Hammer.

• 1973 In Moser’s book [48] the stability problem is raised and some people call the exterior billiard the Moser map.

For literature about billiards see [40, ?, 58].

4.12 Exercices

Exercice 4

Topics: Billiards at a circle

Stability of periodic orbits

1) We study the billiard map in the case, when the billiard table Γ is the circle

γ(q) = (cos(q), sin(q)) .

a) Calculate the generating function h(q, q′) of the billiard map.

b) Calculate p(q, q′) and p′(q, q′).

c) An integral of the map T : X → X is a function F : X ↦ R which satisfies F (T x)) = F (x) for all x = (q, p) ∈ X. Show that p is an integral.

d) Find all periodic points of period k.

e) Calculate the curvature dϕ(q)/dq.

f) Write down the Jacobi matrix L for a periodic orbit with period n.

g) Give an explicit formula for T^{n}(q, p).

h) Make a qualitative picture of the dynamics in the (q, p) coordinates. Make a guess how the dynamics looks, if the curve Γ is a small perturbation of Γ.

2) Given a general billiard table and a periodic orbit of period 2. Give the precice conditions in terms of the curvatures κ_{1}, κ_{2} at the impact points and the distance l between the impact points, such that the orbit is hyperbolic.

Rational maps

The extremely simple dynamical system on C, the quadratic map

f_{c} : z ↦ z^{2} + c

where c is a complex constant is still not completely understood. We will see that each of these maps leaves invariant a set J_{c} ⊂ C, called Julia set such that the dynamics on J_{c} is chaotic and the dynamics on its complement F_{c}, called the Fatou set is tame. It turns out that the parameter space C of such maps is divided into two regions, M, the Mandelbrot set and its complement C \ M, where M can be defined as the set of parameters for which J_{c} is connected.

The quadratic map is not so special as one might think.

Lemma 5.0.1 Every quadratic polynomial R(z) = az^{2}+2bz+d can be conjugated by M(z) = az + b to f(z) = z^{2} + c where c = ad + b − b^{2}.

Proof. Mathematica gives the output True, if running the two lines

R[z_]:=a z^2+2b z+d; M[z_]:=a*z+b; Minv[z_]:=(z-b)/a; c=a*d+b-b^2; F[z_]:=z^2+c;
Simplify[Minv[F[M[z]]]-R[z]]==0

□

Often,especially when dealing with maps on the real line, an other normal form for quadratic maps is chosen:

z ↦ a · z · (1 − z) .

Parametrized like this, the quadratic map is also called the logistic map The linear map M(z) = az − a/2 conjugates z ↦ z^{2} + c with z ↦ z^{2} + c, where c = −a^{2}/4 − a/2.

5.1 Embedding a one-sided shift in the quadratic map

We have seen, how a shift could be embedded in some Henon maps

H : (z, w) ↦ (z^{2} + c − bw, z) ,

where b, c are real parameters. The embedding of the shift worked also when we extended the map to an analytic map on C^{2} by just allowing z, w, b ≠ 0, c to be complex and assuming |c| large enough. In the case b = 0, the dynamics of the Henon map is determined completely by the quadratic map f_{c} : z ↦ z^{2} + c. The map is then no more invertible and we can’t expect to embed an invertible shift in this case. Call X = I^{N} the set of all infinite sequences

X = {x = (x_{1}, x_{2} …) | x_{i} ∈ I}

over a finite alphabet I. The map T (x)_{n} = x_{n+1} is called the one sided shift on X. It is not invertible.

Theorem 5.1.1 There exists a constant M > 0 such that for |c| > M, the quadratic map f_{c} has a shift (X, T) with two symbols embedded.

Proof. The proof is essentially the same as the proof in the Hénon map: Write first the quadratic map in the form

f̃_{a} : q ↦ \frac{q^{2} − 1}{a}

with a = 1/\sqrt{−c}, q = z · a. Consider then the map F : R × C(X, C) → C(X, C) defined by

F (a, q)(x) = a · q(T x) − (q(x)^{2} − 1) .

The map F (0, q) : C(X) → C(X) has the property that every q ∈ C(X, {1, −1}) is a solution of F (0, q) = 0. Take q(x) = x_{0}. The derivative D_{2}F (0, q) is the diagonal operator

D_{2}F (0, q)u = −2q · u

which is invertible. By the implicit function theorem, there exists a solution a ↦ q_{a} = G(a) satisfying F (a, q_{a}) = 0 for small enough a. Define ϕ_{a} : X → R by ϕ_{a}(x) = q(x). Using F (a, q) = 0, we check that

ϕ_{a} ∘ T (x) = q(T x) = \frac{q(x)^{2} − 1}{a} = f̃_{c}(q(x)) = f̃_{c} ∘ ϕ_{a}(x)

for all x ∈ X. The map is injective because if two points x, y are maped into the same point in C then the fact that q_{a}(x) is near to q_{0}(x) = x_{0} implies that x_{0} = y_{0}. Because of the conjugation ϕ_{a} ∘ T^{n}(x) = f̃_{c}^{n} ∘ ϕ_{a}(x) we get also T^{n}(x) = T^{n}(y) and so x_{n} = y_{n} for all n ∈ N. ϕ has a continuous inverse on the image K̃_{a} = ϕ_{a}(X) because every bijective map from a compact metric space to a metric space has a continuous inverse. □

Remarks.

(1) The same proof shows also that the real quadratic map x ↦ x^{2} + c has a one-sided shift embedded if −c ∈ R is large enough.

(2) The dynamics of f̃_{a} on K is conjugated to a one-sided shift. The product measure m on X satisfying m({x | x_{i} = 1}) = m({x | x_{i} = −1) = 1/2 is invariant under T. The measure μ(Y) = m(ϕ_{a}^{−1}(Y)) on C is invariant under f̃_{a} and we have a one sided Bernoulli shift embedded in the quadratic map.

(3) The derivate of f is |f̃′_{a}(q)| = 2q/a. Fix C > 0. For large a and q ∈ K, we have |f̃′_{a}(q)| ≥ C. The Lyapunov exponent λ(q) is growing like − log(a/2) for a → 0.

(4) Look at the two inverses f_{1}(z) = +\sqrt{z − c}, f_{2}(z) = −\sqrt{z − c} of f_{c}. Take the ball Y = B(|c|/2, 0) = {z ∈ C | |z| < |c|/2}. For large |c|, the two maps f_{i} map Y into itself. For z ∈ Y and large |c|, the derivatives f′_{i}(z) = \frac{1}{2·\sqrt{z−c}} satisfy |f′_{i}(z)| ≤ 1/\sqrt{|c|}. The two

maps f_{i} are then contractions on Y with contraction rate ≤ 1/\sqrt{|c|}. In other words, they form an iterated function system and have an unique attractor K_{c}. On the other hand, any point outside Y converges to ∞. There exists therefore a unique attractor K_{c} ⊂ C of the iterated function system f_{i}.

5.2 Special parameter values

We discuss shortly two parameter values c for the quadratic map (see also exercice 5).

1) c = 0. For every point in {|z| < 1}, we have f_{0}^{n}(z) = z^{2^{n}} → 0. Similarly, every point in {|z| > 1} satisfies f_{0}^{n}(z) → ∞. The unit circle {|z| = 1} is invariant and the dynamics on it is

e^{2πiθ} ↦ e^{2πi2θ} .

Looking at the argument of z only, this map can be rewritten as S : θ ↦ 2θ(mod1). We show that S can be conjugated to a shift. Given a sequence (a_{1}, a_{2}, …) ∈ X := {0, 1}^{N}. Define a map ϕ : X ↦ I = [0, 1] by

ϕ(a) = θ = ∑_{n=1}^{∞} a_{n}2^{−n} .

This map is measure preserving but not injective. Denote with m the product measure on X. There exists a set Y ⊂ X with measure one and a set Z ⊂ I of full measure, such that ϕ : Y ↦ Z is invertible. The map ϕ conjugates the shift T on X to the dynamics S on the interval I rsp. the dynamics of f_{0} on the circle {|z| = 1}.

2) c = −2. The map f_{−2} : z ↦ z^{2} − 2 leaves the interval [−2, 2] invariant and is conjugated by a linear transformation to the map z ↦ 4 · z · (1 − z). which preserves the interval [0, 1]. Define the analytic invertible map h(ζ) := ζ +ζ^{−1} from {|z| > 1} to C\[−2, 2]. The identity

f_{−2}(h(ζ)) = h(ζ)^{2} − 2 = ζ^{2} + ζ^{−2} = h(ζ^{2})

gives h^{−1}f_{−2}h = f_{0}. Since the iterates of any |ξ| > 1 converges to ∞ under the dynamics of f_{0}, the same is true for the iterates of any z ∈ C \ [−2, 2] under the iterates of f_{−2}.

5.3 The Mandelbrot set

The set

K_{c} = {z ∈ C | f_{c}^{n}(z) ↛∞}

is called the filled-in Julia set of the quadratic map f_{c}(z) = z^{2} + c. The Mandelbrot set is a subset in the parameter plane defined by

M = {c ∈ C | 0 ∈ K_{c}} = {z ∈ C | f_{c}^{n}(0) ↛∞} .

Here is a little program, which calculates the Mandelbrot set numerically.

M=Compile[{x,y},Module[{z=x+I y,k=0},While[Abs[z]<2.&&k<50,z=z^2+x+I y;++k];k]];
DensityPlot[50-M[x,y],{x,-2.,1.},{y,-1.5,1.5},ColorOutput->RGBColor,PlotPoints->200,Mesh->False];

The program uses the property that the Mandelbrot set is the set of parameter values c ∈ C, where f_{c}^{n}(0) stays bounded. One computes the orbit of 0 and lookes, how many iterations M(z) are needed to get into the set {|z| > 2}. A point is in the Mandelbrot set, if and only if M(c) = ∞. After an idea of Hubbard, the program puts colors at each point c according to the value of M(c). We will see that 1/M(c) measures somehow the distance to M.

5.4 The Julia sets

Given an open set U ⊂ C. A function f is analytic (or holomorphic) on U, if it is complex differentiable at each point of U. A function f is meromorphic on U, if it is analytic in U except for poles or if f(z) = ∞ for all z ∈ U.

We say, a sequence of analytic maps f_{n} converges uniformly to f on a compact set K ⊂ U, if f_{n} → f in C(K), which means max_{x∈K} |f_{n}(x) − f(x)| → 0. A family of analytic maps F is called normal, if every sequence f_{n} ∈ F contains a subsequence which converges uniformly on any compact subset of U. The limit function f does not necessarily have to be in F.

Remark. The topology of convergence on compact subsets of the set of analytic maps on U can be given by a metric. Normality is just praecompactness in this topology. This means that a family F is normal, if and only if its closure is compact with respect to this metric. The theorem of Arzela-Ascoli (see Alfors p. 224), says that a family F of analytic functions on U is normal, if and only if the functions are equicontinuous on every compact set K ⊂ U and if for every z ∈ U, the set {f(z) | f ∈ F} is bounded.

It is useful to consider the dynamics of an analytic map on the Riemann sphere \overline{C} = C∪\{∞\} which is an analytic manifold (Riemann surface) homeomorphic to the two-dimensional sphere. Given an open subset U of \overline{C}. A family F of meromorphic functions f : U ↦ \overline{C} is called normal, if every sequence f_{n} ∈ F contains a subsequence which converges uniformly on compact subset of U.

The theorem of Arzela-Ascoli implies that a family F of meromorphic functions on U ⊂ \overline{C} is normal, if and only if it is equicontinuous on every compact set K ⊂ U.

Given a meromorphic map f : \overline{C} ↦ \overline{C}. A point z ∈ C is an element of the Fatou set F_{f} of f, if the set {f^{n}}_{n∈N} is normal in some neighborhood of z. The Julia set J_{f} is defined to be the complement of the Fatou set.

Here are some examples of filled-in Julia sets computed with a similar algorithm as the Mandelbrot set. The Julia sets are the boundaries of these filled in Julia sets.

J=Compile[{x,y,u,v},
  Module[{z=u+I v,k=0},While[Abs[z]<200.&&k<50,z=z^2+x+I y;++k];k]];
Julia[x_,y_]:=DensityPlot[50-J[x,y,u,v],{u,-1.,1.},{v,-1.,1.},
  PlotPoints->250,Mesh->False];
DouadyRabit:=Julia[-0.121,0.739];           Dragon:=Julia[0.36,0.1]
Gullik:=Julia[0.32,0.043];                  Dendrite:=Julia[0,1]
Airplane:=Julia[-1.754878,0.];              Cauliflower:=Julia[0.25,0.]
Dust1:=Julia[0.255,0]                       Dust2:=Julia[-0.75,0.1]

5.5 Preliminaries

A topological space X is connected, if the only sets which are both open and closed are ∅ and X. A subset Y ⊂ X is connected, if it is connected with respect to the induced topology. An open subset of a topological space is therefore connected if and only if it can’t be written as the disjoint union of two open nonempty sets and a closed set is connected if and only if it can’t be written as the disjoint union of two closed nonempty sets. An open set Y of C is connected, if and only if two points can be connected by a polygon lying completely inside Y A closed subset of \overline{C} = C ∪ \{∞\} is disconnected if and only if there exists a Jordan curve which separates the set. A nonempty connected open subset of C is also called a region or a domain. A subset Y ⊂ X is called simply connected, if any closed curve in Y can be contracted in Y continuously to a point.

Lemma 5.5.1 Given an open subset of D ⊂ \overline{C}. The complement is connected if and only if each component of D is simply connected.

Proof. (informal) D is not simply connected then one can find a curve γ in D which is not contractible. This curve separates the complement of D onto two not connected sets. If \overline{C} \ D is not connected then it can be devided by a closed curve into two components. This closed curve is not contractible. □

5.6 The connectivity of the Mandelbrot set

It show first that M is simply connected. In the proof, we use the Maximum principle, which is one of the of the most useful general theorems in complex analysis.

Lemma 5.6.1 (Maximum principle) If f is analytic and not constant in a domain U, then |f(z)| has no maximum in U.

Proof. Assume z_{0} ∈ U is a maximum of |f(z)|. Cauchy’s integral formula gives for r < dist(z_{0}, δU)

|f(z_{0})| ≤ \frac{1}{2π} ∫_{0}^{2π} |f(z_{0} + re^{i}θ)| dθ .

Because |f(z_{0})| ≥ |f(z_{0} + re^{i}θ)|, the function is constant on the circle |f(z_{0} + re^{i}θ)| and so on the set K_{1} = {|z − z_{0}| ≤ dist(z_{0}, δU)}. Take any point z_{1} in the boundary of K_{1}. The same argument as before shows that f is constant on K_{2} = {|z − z_{0}| ≤ dist(z_{0}, δU)}. We can sucessively cover the whole set U by closed sets K_{n} on which f is constant. This implies f is a constant function. □

Theorem 5.6.2 M is a closed, simply connected subset of the disc \overline{D}_{2} = \{|z| ≤ 2\}. Moreover M = {z | |f_{c}^{n}(0)| ≤ 2, ∀n ∈ N}.

Proof. If |c| > 2, then one shows by induction that

|f_{c}^{n+1}(0)| ≥ |c|(|c| − 1)^{2^{n−1}}, n ≥ 1 ,

so that c ∉ M. Therefore M ⊂ D_{2}. If |f_{c}^{m}(0)| = 2 + δ for some m, one obtains |f_{c}^{n+k}(0)| ≥ 2 + 4^{k}δ and again c ∉ M. M is therefore contained in

M̃ = ⋂_{n} M_{n}

with M_{n} = f^{−n}(D_{2}). On the other hand, M̃ ⊂ M by definition and so M = M̃. M is closed as the intersection of closed sets M_{n}. The maximum principle implies that each of the sets \overline{C} \ M_{n} is connected because the boundary of such a set is the set, where |f^{n}| takes its minimum 2. Because for any sequence D_{n} ⊂ D_{n+1} of connected domains, the union is also simply connected, the union ⋃_{n} C \ M_{n} is also connected. By Lemma 5.5.1, M̃ = M is simply connected. □

Our aim is to prove the following theorem.

Theorem 5.6.3 (Douady-Hubbard) M is connected. More precicely, there exists a conformal isomomorphism Φ : \overline{C} \ M → \overline{C} \ D.

Remark. It is the big unsolved problem in the field whether M is locally connected (a topological space is locally connedtd if any neighborhood of a point contains a connected neighborhood.)

5.7 Local dynamics near fixed points

Let z_{0} be a fixed point of an analytic map f. It is called attractive, if |f′(z_{0})| < 1, repelling, if |f′(z_{0})| > 1 and superattractive, if f′(z_{0}) = 0. Near a superattractive fixed point z_{0}, the map can be written as f(z) = ∑_{n=k}^{∞} a_{n}(z − z_{0})^{n} with a_{k} ≠ 0 and k ≥ 2. The next lemma which is due to Böttcher sais that there exist coordinates near z_{0} such that the map looks locally very simple near a superattractive fixed point. We can assume after a translation z_{0} = 0. We also assume a_{k} = 1.

Lemma 5.7.1 (Böttcher-Fatou) Let f(z) = z^{k} +a_{k+1}z^{k+1} +… with k ≥ 2 be an analytic map in a neighborhood

of 0. Define ϕ_{n}(z) = (f^{n}(z))^{1/k^{n}} = z + O(z^{2}). Then, there exists a neighbor-
hood U of 0 and an analytic diffeomorphism ϕ = lim_{n→∞} ϕ_{n}(z) : U → B_{r}(0)
such that ϕ ∘ f ∘ ϕ^{−1}(z) = z^{k}, ϕ(0) = 0 and ϕ′(0) = 1.

Proof. We have only to show the uniform convergence of ϕ_{n}. The properties ϕ(f(z)) = ϕ(z)^{k} as well as ϕ(0) = 0 and ϕ′(0) = 1 follow from the formula. The function

h(z) := log(\frac{f(z)^{1/k}}{z})

with the chosen root f(z))^{1/k} = z + O(z^{2}) is holomorphic in a neighborhood U of 0 and there exists a constant C such that |h(z)| ≤ C|z| for z ∈ U. U can be chosen so small that f(U) ⊂ U and |f(z)| ≤ |z|. We can write ϕ(z) as an infinite product

ϕ(z) = z · \frac{ϕ_{1}(z)}{z} · \frac{ϕ_{2}(z)}{ϕ_{1}(z)} · \frac{ϕ_{3}(z)}{ϕ_{2}(z)} ….

This product converges, because ∑_{n=0}^{∞} log \frac{ϕ_{n+1}(z)}{ϕ_{n}(z)} converges absolutely and uniformly for z ∈ U:

| log \frac{ϕ_{n+1}(z)}{ϕ_{n}(z)}| = | log [\frac{(f ∘ f^{n}(z))^{1/k}}{f^{n}(z)} ]^{1/k^{n}} | = \frac{1}{k^{n}} · |h(f^{n}(z))| ≤ \frac{1}{k^{n}} C · |f^{n}(z)| ≤ \frac{C · |z|}{k^{n}} .

□

Corollary 5.7.2 If c ↦ f_{c}(z) is a family of analytic maps such that c ↦ f_{c}(z) is analytic for fixed z, and c is in a compact subset of C, then the map (c, z) ↦ ϕ_{c}(z) is analytic in two variables.

Proof. Use the same estimates as in the previous proof: the maps (c, z) ↦ ϕ_{n}(c, z) are analytic and the infinite product converges absolutely and uniformly on a neighborhood U of 0. □

Corollary 5.7.3 The Julia set J_{c} is a compact nonempty set.

Proof. (i) The Julia set is bounded: the Lemma of Boettcher-Fatou implies that every point z with large enough |z| is converging to ∞. This means that a whole neighborhood U of z is converging to ∞. In other words, the family F = {f_{c}^{n}}_{n∈N} is normal, because every sequence in F converges to the constant function ∞. (ii) The Julia set is closed: this follows from the definition, because the Fatou set F_{c} is open. (iii) Assume the Julia set is empty. The family F = \{f_{c}^{n}\} is then normal on \overline{C}. This means that for any sequence f_{n} in F, there is a subsequence f_{n_{k}} converging to an analytic function f : \overline{C} → \overline{C}. Because such a function can have only finitely many zeros and poles, it must be a rational function P/Q where P, Q are polynomials. If f_{n_{k}} → f, there are eventually the same number of zeros of f_{n_{k}} and f. But the number of zeros of f_{n_{k}} (counted with multiplicity) is growing monotonically. This contradition shows that the assumption J_{c} = ∅ was wrong. □

Corollary 5.7.4 The Julia set J_{c} is contained in the filled in Julia set K_{c}. The later is the union of J_{c} and the bounded components of F_{c}.

Proof. Because J_{c} is bounded and f invariant, every orbit starting in J_{c} is bounded and belongs by definition to the filled-in Julia set. If a point is in a bounded component of F_{c}, its forward orbit stays bounded and it belongs to the filled in Julia set. On the other hand, if a point is not in the Julia set or a bounded component of F_{c}, then it belongs to an unbounded component of F_{c}. □

5.8 The Green function

A continuous function G : C ↦ R is called the potential theoretical Green function of a compact set K ⊂ C, if G is harmonic outside K, vanishing on C \ K and has the property that G(z) − log(z) is bounded near z = ∞.

Proposition 5.8.1 The Green function G_{c} exists for the filled-in Julia set K_{c} of the polynomial f_{c}. The map (z, c) ↦ G_{c}(z) is continuous.

Proof. Lemma 5.7.1 assures the existence of the function ϕ_{c} conjugating f_{c} with z ↦ z^{2} in a neighborhood U_{c} of ∞. Define for z ∈ U_{c}

G_{c}(z) = log |ϕ_{c}(z)| .

This function is harmonic in U_{c} and growing like log |z| because by Boettcher satisfies |f_{c}^{n}(z)| ≥ C|z|^{2^{n}} for some constant C and so

G_{c}(z) = lim_{n→∞} \frac{1}{2^{n}} log |f_{c}^{n}(z)| .

Although G_{c} is only defined in U_{c}, there is one and only one extension to all of C which is continuous and satisfies

G_{c}(z) = G_{c}(f_{c}(z))/2 .  (5.1)

In fact, we define G_{c}(z) = 0 for z ∈ K_{c}, and G_{c}(z) = G(f_{c}^{n}(z))/2^{n} otherwise, where n is large enough so that f_{c}^{n}(z) ∈ U. We know from this extension that G_{c} is a smooth real analytic function outside K_{c}. From the maximum principle, we know that G_{c}(z) > 0 for z ∈ C \ K_{c}. We have still to show that G_{c} is continuous in order to see that it is the Green function. The continuity follows from the stronger statement:

(z, c) ↦ G_{c}(z) is jointly continuous. G_{c}^{−1}([0, ϵ)) is open in C^{2} for all ϵ > 0 if and only if there exists n such that

A_{n} := {(c, z) | G_{c}(f_{c}^{n}(z)) ≥ 2^{n}ϵ}

is closed ∀ϵ > 0. Given r > 0. There exists a ball of radius b which contains all the sets K_{c} for |c| ≤ r. For R ≥ G_{r}(b), all the solutions ξ of G_{c}(ξ) ≥ R satisfy |ξ| ≥ b if |c| ≤ r. The set B = {(c, ξ) | G_{c}(ξ ≥ R} ∩ {|c| ≤ r} is closed. For n large enough, also A_{n} ∩ {|c| ≤ r} is closed and A_{n} is closed. □

5.9 M is connected

Proof. The Böttcher function ϕ_{c}(z) can be extended to

S_{c} := {z | G_{c}(z) > G_{c}(0)} .

Continue defining ϕ_{c}(z) := \sqrt{ϕ_{c}(z^{2} + c)} to get ϕ_{c} having defined in larger and larger regions. This can be done as long as the region ϕ_{c}^{−1}({r}) is connected (this assures that the derivative of ϕ_{c} is not vanishing). Because Equation 5.1 gives G_{c}(c) = 2G_{c}(0) > G_{c}(0), every c is contained in the set S_{c} and the map

Φ : c ↦ G_{c}(c)

is well defined. It is holomorphic outside M and can be written as

Φ(z) = lim_{n→∞}[f_{c}^{n}(c)]^{1/2^{n}} .

Claim:

Φ : \overline{C} \ M → \overline{C} \ \overline{D}

is an analytic diffeomorphism. (This implies that the complement of M is simply connected in \overline{C}, which is equivalent with the fact that M is connected).

(1) Φ is analytic outside M. This follows from Corollary 5.7.2.

(2) For c_{n} → M, we have |Φ(c_{n})| → 1. Proof. Continuity of the Green function.

(3) The map Φ is proper. (A map is called proper if the inverse of any compact set is compact). Given a compact set K ⊂ C \ D. The two compacta D and and K have positive distance. Assume ϕ^{−1}(K) is not compact. Then, there exists a sequence c_{n} ∈ Φ^{−1}(K) with c_{n} → c_{0} ∈ M so that |Φ(c_{n})| → 1. This is not possible because Φ(c_{n}) ∈ K is bounded away from D.

(4) The map Φ is open (it maps open sets into open sets). This follows from the fact that Φ is holomorphic. (This is called open mapping theorem).

(5) The map Φ maps closed sets into closed sets. A proper, continuous map Φ : X → Y between two locally compact metric spaces X, Y has this property. Proof. Given a closed set A ⊂ X. Take a sequence Φ(a_{n}) in Φ(A) which converges to b ∈ Y. Take a compact neighborhood K of b (use local compactness of Y). Then Φ^{−1}(K ∩ \overline{ϕ(A)}) is compact and contains almost all a_{n}. The sequence a_{n} contains therefore an accumulation point a ∈ X. The continuity implies Φ(a_{n}) → Φ(a) = b for a subsequence. Therefore b ∈ Φ(K) and Φ(K) is closed.

(6) Φ is surjective. The image of Φ(\overline{C} \ M) is an open subset of set \overline{C} \ \overline{D} because Φ is open. The image of the boundary of M is (use (5)) a closed subset of \overline{C} \ D which coincides with the boundary of D because by Proposition 5.8.1 G_{c}(c) → 0 as c → M.

(7) Φ is injective. Because the map Φ is proper, the inverse image of a point s is finite. There exists therefore a curve Γ enclosing all the points of Φ^{−1}(s). Denote with ♯A the number of elements in a set A. By the argument principle (see Alfors p. 152), we have

♯(ϕ^{−1}(s)) = \frac{1}{2πi} ∫_{Γ} \frac{Φ′(z)}{Φ(z) − s} dz

and this number is locally constant. Given M > 0, we can find a curve Γ which works simultaneously for all |s| ≤ M. Because Φ is surjective and ♯(ϕ^{−1}(∞)) = 1, we get that ♯(ϕ^{−1}(s)) = 1 for all z ∈ C \ \overline{D} and ϕ is injective.

(8) The map Φ^{−1} exists on C \ D and is analytic. Because an injective differentiable open map has a differentiable inverse, (this is also called Goursat’s theorem) the inverse is analytic. □

5.10 The Mandelbar set and other sets

The Mandelbar set or tricorn is defined as

\overline{M} := \{c ∈ C | T_{c}^{n}(0) ↛∞\} .

Similiarily as the Mandelbrot set M contains the cardioid {w −w^{2} | |w| ≤ 1/2}, the Mandelbar set \overline{M} contains the deltoid \{w −\overline{w}^{2} | |w| ≤ 1/2\} which is already a rough approximation of the shape of the Mandelbar set. The Mandelbar set is symmetric under a rotation about the origin by an angle ω = exp(2πi/3) = \overline{ω}^{2} because

T_{ωc}(z) = \overline{z}^{2} + ωc = ω((\overline{ωz})^{2} + c = T_{c}(ωz) .

It is also symmetric with respect to conjugation z ↦ \overline{z} so that the tringle group D_{3} is a symmetry group of \overline{M}. Both sets M and \overline{M} are compact and subsets of \{|z| ≤ 2\}. Indeed they can be charaterized as the intersecion of the nested list of compact sets {c || |T_{c}^{n}(c)| ≤ 2}. The proof of this goes like in the case of the Mandelbrot set: Assume |c| > 2. Since

|T_{c}^{n+1}(c)| = |\overline{T_{c}^{n}(c)}^{2} + c| ≥ |\overline{T_{c}^{n}(c)}|^{2} − |c|

we deduce by induction that |T_{c}^{n}(c)| > |c| and α_{n} = |T_{c}^{n}(c)/c| satisfies

α_{n+1} > 2α_{n} − 1

which is going to infinity. We have shown that \overline{M} ⊂ \{|z| ≤ 2\}. We see also that \overline{M} ⊂ {c | |T_{c}^{n}(c)| ≤ 2} because if c ∈ M and α_{n} = |T_{c}^{n}(c)| > 2 for some n then α_{n} → ∞. Nakane has shown that like the Mandelbrot set, the tricorn is connected [49]. Since the parameterdependence of the map T_{c} is not analytic, the proof does not go through. The strategy of the proof is however similar: the Böttcher function

ϕ_{c}(z) = lim_{n→∞}(T_{c}^{2n}(z))^{2^{−2n}}

(the root is chosen such that it is positive on the positive real axis) conjutates the map T_{c} with T_{0} : z ↦ \overline{z}^{2}

ϕ_{c}T_{c}ϕ_{c}^{−1} = T_{0}

near infinity and ϕ_{c} is a conformal isomorphism of

\overline{C} \ K_{c} → \overline{C} \ \overline{D}

where K_{c} is the filled-in Julia set

K_{c} + {z | lim_{n→∞} |T_{c}^{n}(z)| ≠ ∞}

which is the set of parameters such that the boundary J_{c} of K_{c} is connected. Also here, J_{c} is the set of points, on which T_{c}^{n} is not normal or the closure of the repelling periodic orbits. Define again

Φ(c) = ϕ_{c}(c) .

Nakane shows that the map Φ is a real-analytic diffeomorphism

\overline{C} \ \overline{M} → \overline{C} \ \overline{D}

which implies as a corollary that \overline{M} is connected. One again shows that the map Φ is proper and locally invertible so that it is then a covering map. Since \overline{C} \ \overline{D} is simply conneced and the degree is one at ∞ the covering is trivial. Since Φ is real analytic, its inverse is also real analytic.

Other sets in Parameter spaces one can get many sets analoguous to the Mandelbrot set. Some people go further: Using iteration of purely mathematical fucntions, Cliff Pickover of IBM created a lab dish of ”biomorphs”. (crf. [?])

The family z ↦ z^{3} + c.

M=Compile[{x,y},Module[{z=x+I y,k=0},While[Abs[z]<2.&&k<50,z=z^3+x+I y;++k];k]];
DensityPlot[50-M[x,y],{x,-1.,1.},{y,-1.5,1.5},PlotPoints->200,Mesh->False];

The family z ↦ c · cos(z).

M=Compile[{x,y},
  Block[{z=x+I y,k=0},While[Abs[z]<9&&k<50,z=(x+I y)*Cos[z];++k];k]];
DensityPlot[50-M[x,y],{x,-3.,3.},{y,-3.5,3.5},PlotPoints->200,Mesh->False];

5.11 Iteration of rational maps

Given a rational map

R : \overline{C} → \overline{C}, z ↦ \frac{P (z)}{Q(z)} ,

where P, Q are polynomials with no common factor. The degree d = d(R) of R is the maximum of the degrees of P and Q. We assume aways that the degree of a rational map is ≥ 2. (degree one maps are just Möbius transformations.) A critical point of R is a point z of the sphere \overline{C} where R is not locally bijective. The image of a critical point is called a critical value. The order of a critical point is the integer m such that R is (m + 1) to 1 in a punctured neighborhood of z. Critical points consist of solutions of R′(z) = 0 and of poles of R of order two or higher. There are clearly only finitely many critical points.

Lemma 5.11.1 There are 2d + 1 critical points if R, Proof. Composing R with a Möbius transformation does not change the number of critical points. By doing so, we can assume without loss of generality that ∞ is neither a critical point nor a critical value and that R(∞) = 0. We can write

R(z) = \frac{P (z)}{Q(z)} = α\frac{z^{d−1} + …}{z^{d} + …}

and get

R′(z) = \frac{Q(z)P′(z) − Q′(z)P (z)}{Q(z)^{2}} = −α\frac{z^{2d−2} + …}{Q(z)^{2}}

so that there are 2d − 2 critical values because R has by assumption no poles of order 2 or higher. □

The Fatou set F (R) is defined to be the set of points z ∈ \overline{C} such that the family \{R^{n}\}_{n} ∈ N is normal in a neighborhood of z. The Julia set is the complement of the Fatou set. By Arzela’s theorem, a family of analytic functions on \overline{C} is normal on an open set U ⊂ \overline{C} if and only if it is equicontinuous on U with respect to the spherical metric on \overline{C}. The Fatou set is open and the Julia set compact.

Lemma 5.11.2 The Julia set of a rational map with degree ≥ 2 is nonempty.

Proof. Suppose that the Julia set is empty. The \{R^{n}\} is a normal family on \overline{C} and there is a subsequence n_{j} → ∞ such that R^{n_{j}}(z) → f(z) for some analytic function \overline{C} → \overline{C}. Since f is analytic on all \overline{C}, it is rational.

Claim. Assume a sequence of rational maps f_{n} converges to a rational map f, then the degrees d(f_{n}) either converge to d(f) or d(f_{n}) → 0 which means that f is a constant. Proof. Assume f_{n} converges not to a constant. Then, f has some zeros and poles. Recall the argument principle which says for a given closed curve γ avoiding the zeros and poles of a meromorphic function f, we have

\frac{1}{2πi} ∫_{γ} \frac{f′(z)}{f(z)} dz = ∑_{f(a)=0} n(a) − ∑_{f(b)=∞} n(b) .

Take a closed curve γ around a zero or pole of f. Because f_{n} converges to f, we know that f_{n} has eventually the same number of zeros and poles as f. Therefore the degrees converge.

Because d(R_{n}) = d(R)^{n} → ∞, the just made remark implies that f must be a constant. Therefore, the values of R^{n_{j}} are eventuelly in a small neighborhood of the constant f which is not possible because R^{n} covers \overline{C}. □

Remark. Rational maps of degree 1 are Möbius transformations. They can have an empty Julia set as the example f(z) = z shows.

A set E is completely invariant if both E and its complement are invariant. Because R is a surjective map, this is equivalent with R^{−1}(E) = E.

Theorem 5.11.3 The Julia set J is completely invariant and is identical to the Julia set of R^{N} for any N ≥ 1.

Proof. By definition R^{−1}(F) ⊂ F. To show the converse assume z ∈ F. There exists a sequence n_{j} such that R^{n_{j}} ∘R converges uniformly on a neighborhood U of z. Because R(U) is open (see Alfors p. 132) and a neighborhood of R(z) and R^{n_{j}} converges uniformly on R(U), also R(z) ∈ F. Because {R^{n}} is normal if and only if {R^{nN}} is normal, the Fatou set of R^{n} coincides with the Fatou set of R. □

A family of analytic functions F on the unit disc, bounded by some constant is normal, because the Cauchy estimate |f′(z)| ≤ M/(1 − r) for any |z| < r < 1 implies that F is equicontinuous on any compact subset. Covering an arbitrary region by discs shows that a a uniformly bounded family of functions on this region is normal. The following criterium for normality is very important.

Theorem 5.11.4 (Montel) If a family F of meromorphic functions ommiting three values, then F is normal.

Proof. (informal) On shows first that the thrice-punctured sphere has a uniform cover which is conformally equivalent to the open unit disc. The family F can then be lifted to a family of conformal maps having values in the unit disc. With Arzela-Ascoli one concludes that F is normal. □

Given a neighborhood U of a point z ∈ J(R). By Montel’s theorem, the family {R^{n}} on any such U omits a set E_{z} containing at most two points.

Lemma 5.11.5 The set E := E_{z} is independent of z and contained in the Fatou set.

Proof. By definition R^{−1}(E_{z}) ⊂ E_{z}. We first show that E_{z} is independent of z. Case (1): E_{z} = {a}. Conjugate R with a Möbius transformation to get a = ∞. Because only z = ∞ can be mapped to ∞, R is a polynomial and clearly E_{z} is independent of z. Case (2): E_{z} = {a, b}. We can conjugate R to get {a, b} = {0, ∞}. Then eigher R(0) = 0, R(∞) = ∞ or R(0) = ∞, R(∞) = 0. In the first case R is a polynomial with the only zero 0 and R(z) = cz^{n}. In the second case, we consider R̃(z) = 1/R(z). The same argument shows that R̃(z) = c · z^{n} and so R(z) = 1/c · z^{−n}. In any case, the set E_{z} is independent of z. Case (0): E_{z} = ∅ is also independent of z because the other cases are. Assume z ∈ E is in the Julia set. The set E = E_{z} does not contain z. Contradiction. □

The set E is called the exceptional set of R. Call {R^{−n}(z) | n ≥ 0} the inverse orbit of a point z ∈ \overline{C}. The α-limit set of z is the set of accumulation points of the inverse orbit of z. If a set does not contain isolated points, then it is called perfect.

Corollary 5.11.6 a) The α limit set of any point z outside E is dense in J.

b) The inverse orbit of any point z ∈ J is dense in J.

c) Any nonempty completely invariant subset of J is dense in J.

d) Let D be a union of components of F that is completely invariant, then J = δD.

e) The Julia set is perfect.

Proof. a) Given any point w ∈ J and a neighborhood U of w. Because ⋃_{n} R^{n}(U) = \overline{C} \ E, there exists n such that R^{−n}(z) ∈ U.

b) follows from a) and the fact that J is completely invariant.

c) follows from b). Take any point z in the completely invarinant set. Because the inverse orbit is dense, the set must also be dense.

d) The boundary δ(D) is a completely invariant subset of J. Use c).

e) Given z_{0} ∈ J and an open neighborhood U of z_{0}. 1.Case: Assume z_{0} is not periodic. Since by b), the inverse orbit of z_{0} is dense and contained in J, there exists w ∈ U ∩ J and n ∈ N such that T^{n}(w) = z_{0}. 2.Case: Let n be minimal satisfying R^{n}(z_{0}) = z_{0}. If z_{0} would be the only solution of R^{n}(z) = z_{0}, then z_{0} would be a superattracting fixed point for R^{n}, contradicting z_{0} ∈ J. (We use here the local description of superattractive, attractive or repelling fixed points.) There exists therefore z_{1} ≠ z_{0} with R^{n}(z_{1}) = z_{0} and n is the smallest number satisfying this. Since z_{1} is not periodic, z_{1} has a preimage in U ∩ J different from z_{0}. □

The corollary b) contains the key to compute J numerically by the inverse iteration method. Notice however that the α limit set is not necessarily contained in J. We could be unlucky and start with a periodic orbit in F.

The basin of attraction of an attracting fixed point z_{0} is the set of points z such that R^{n}(z) → z_{0} for n → ∞. The baisin of attraction of an attractive periodic orbit z_{0}, … is the basin of attraction of the fixed point of R^{m}(z_{0}) = z_{0}. The immediate basin of attraction is the union of the connected components containing a point of the periodic orbit.

Lemma 5.11.7 If z_{0} is an attracting periodic point, the basin of attracion is a union of components of the Fatou set. The boundary of this basin is the Julia set.

Proof. Take an open neighborhood of the periodic orbit of z_{0} which is contained in the Fatou set. The union A of the backward iterates of U is an open subset of the Fatou set. Any z_{1} at the boundary of this set is in the Julia set: let V be a neighborhood of z_{1}. Any point z ∈ V ∩ A is converging to the periodic orbit wheras any point z ∈ V \ A remains outside A. The family R^{n} can not be normal on V. Since A is completely invariant, the boundary of A is a completely invariant set of J which must coincide with J. □

Proposition 5.11.8 If z_{0} is an attracting periodic orbit. The immediate basin of attraction A of z_{0} contains at least one critical point.

Proof. We assume first that z_{0} is a fixed point. If z_{0} is superattractive then it is itself a critical point. We assume therefore that the R′(z_{0}) = λ satisfies 0 < |λ| < 1. Let U_{0} be a small disk which is invariant under R and on which the analytic branch f of R^{−1} satisfying f(z_{0}) = z_{0} is defined. The branch f is injective and maps U_{0} onto a simply connected set U_{1} satisfying U_{0} ⊂ U_{1} ⊂ A. Iterating this we obtain a sequence U_{n+1} = f(U_{n}) ⊃ U_{n}. Assume we can extend f analytically to all the sets U_{n}. The family f^{n}U_{0} → U_{n} is then a family omitting J. By Montel, such a family is normal. But this is impossible since z_{0} is a repelling fixed point. This shows that there is a U_{n} to which we cannot extend f. There exists then a critical point w ∈ A such that R(p) ∈ U_{n}. If z_{0} is periodic with period n > 1, the argument shows that each component of A contains a critical point of R^{n}. Since (R^{n})′(w) = ∏ R′(R^{j}(w)), there must also exist a critical point w of R. □

In order to avoid a complete discussion about local dynamics around fixed points, we assume, the following theorem to be true.

Theorem 5.11.9 (Shishikura) There is an analytic permutation of a rational map such that all neutral cycles become attractive.

Corollary 5.11.10 The total number of attracting and neutral cycles (counting with multiplicity) is at most 2d − 2.

Proof. Perturb the map so that it has maximal 2d − d attractive cycles according to the theorem of Shishikura. □

Theorem 5.11.11

The Julia set J is the closure of the repelling periodic points.

Proof. Assume there exists an open disc that meets J and that contains no periodic points. By eventuelly choosing a smaller disc we can assume that U contains neither poles nor critical values of R. Take two branches f_{1}, f_{2} of R^{−1} on U. Since there by assumption no solutions of R^{m}(z) = z in U,

g_{n} = \frac{R^{n} − f_{1}}{R^{n} − f_{2}} · \frac{z − f_{2}}{z − f_{1}}

omits the values 0, 1, ∞ in U. By Montel’s theorem, {g_{n}} is normal. Because we can solve for R^{n} also R^{n} is normal. This is a contradiction. It follows that our assumption about periodic orbits is wrong and there exists a dense set of peridic orbits. Because there are only finitely many repelling or neutral orbits and J does not have isolated points, the repelling orbits are dense. □

5.12 Embedding of a one-sided shift in rational maps

Given a rational map f : z ↦ γ · \frac{P (z)}{Q(z)}, where P and Q are polynomial and γ is a parameter. Call I and J the set of simple zeros of P and Q. Denote with X_{I} = I^{N} rsp. X_{J} = J^{N} the sets of one-sided sequences with alphabet I rsp. J and with T the shift on X_{I} or X_{J}. To embed (X_{I}, T) in the rational map f, we look for solutions q ∈ C(X) of

Q(q) · q(T) − γ · P (q) = 0 .

For γ = 0, any q ∈ C(X, I) is a solution, for γ = ∞, (devide the equation by γ and make the limit γ → ∞), any q ∈ C(X, J) is a solution. In both cases, the implicit function theorem allows to extend the solution to values near γ = 0 or γ = ∞:

Proposition 5.12.1 For large enough |γ|, a Bernoulli shift (X_{I}, T) can be embedded into the rational map f, for small enough |γ|, a Bernoulli shift (X_{J}, T) can be embedded into the rational map f.

5.13 Epilog

Why studiing the quadratic map and rational maps?

• There are wonderful pictures to see while studding such dynamical systems.

• The quadratic map is the simplest nonlinar map one can think of.

• There are still many open problems.

• Various subjects in mathematics are involved: complex function theory, potential theory, plane topology, ergodic theory

• Julia sets of rational maps arrise also in other contexts like Schroedinger operators or statistical mechanics.

The following books give introductions in complex dynamics: [4, 11, 13, 47, 45]. We recommend also the overview articles [8, 9, 21].

5.14 Exercices and open research problems

The main question is: is M locally connected? By Carathéodory’s theorem equivalent with the problem to extend Φ^{−1} continuously to a map from δD onto δM. The result would follow if there were some uniform estimate of the escape times for c in terms of its distance to M (see Carleson p. 140).

5.15 Exercices

Exercice 5

Topics: Julia and Mandelbrot sets

The Mandelbrot set.

1) Show that the intersection of M with R is the interval [−2, 1/4].

Hint: Proceed in three steps:

(i) show that M ⊂ {|z| ≤ 2}

(ii) show that for c > 1/4, f_{c}^{n}(0) grows monotonically and has no limit.

(iii) denote with a the larger root of f_{c}(x) = x, show for every c ∈ [−2, 1/4] satisfies |f_{c}(0)| ≤ a, |f_{c}(0)^{n}| ≤ a ⇒ |f_{c}(0)^{n+1}| ≤ a.

2) Define for any a ∈ R the ring R_{a} = R[\sqrt{a}] as the set of pairs x + \sqrt{a}y. The addition is

defined as

x_{1}\sqrt{a}y_{1} + x_{2} + \sqrt{a}y_{2} = (x_{1} + y_{1}) + \sqrt{a}(x_{2} + y_{2})

the multiplication as

(x_{1} + \sqrt{a}y_{1})(x_{2} + \sqrt{a})y_{2} = (x_{1}y_{1} + ax_{1}y_{1}) + \sqrt{a}(x_{1}y_{2} + x_{2}y_{1}) .

An example is the complex plane R_{−1} = R[\sqrt{−1}]. Given c ∈ R_{a}. The quadratic map f_{c} in R_{a} is defined as f_{c}(z) = z^{2} + c. Define in the ring R_{a} the Mandelbrot set M_{a} as the complement of the set of points c ∈ R_{a} such that f_{c}^{n}(0) → ∞.

a) Show that the Mandelbrot set M_{−2} is homeomorphic to M = M_{−1}.

b) Compute explicitly the Mandelbrot set M_{0}.

c) Compute explicitly the Mandelbrot set M_{1}.

d) Determine all Mandelbrot sets M_{a} for all a > 0.

The Julia sets.

3) a) Compute the Julia sets in the case c = 0.

b) Compute the Julia set in the case c = −2.

For fun.

4) Verify the following statement of Douady about the map f_{c}. ”A point |z| > |c| + 2 has an image f_{c}^{12}(z) so far away that the modulus of this number is larger then the ratio of the volume of the known universe to the volume of a proton.” (Hint: Length of universe 10^{26}m, length of proton 10^{−15}m).

Standard maps

6.1 The standard map

The Chiricov or standard mapping on the torus T^{2} = R^{2}/(2πZ)^{2} is defined as

T̃_{γ} : ( \begin{matrix}x \\ y \end{matrix} ) ↦ ( \begin{matrix}x + y + γ sin(x) \\ y + γ sin(x) \end{matrix} )

with real parameter γ. The map is defined on the compact metric space T^{2}. This is opposite to the Hénon maps which were defined on R^{2} A linear involutive change of coordinates (x, y) ↦ (q, p) = (−x, y − x) brings the map T into the form

T_{γ} : ( \begin{matrix}q \\ p \end{matrix} ) ↦ ( \begin{matrix}−p + 2q + γ sin(q) \\ q \end{matrix} ) .

An orbit (q_{n}, p_{n}) of the standard map is determined by the second order recurrence relation

q_{n+1} − 2q_{n} + q_{n−1} + γ sin(q_{n}) = 0 .

Here is a little film done as follows: each picture shows an orbit with the same initial condition. The parameter γ is increased in each step.

T[{q_,p_},g_]:=N[Mod[{2q-p+g*Sin[q],q},2 Pi]];
Orb[p0_,n_,g_]:=Module[{},S[p_]:=T[p,g];NestList[S,p0,n-1]];

Pict[p0_,n_,g_]:=ListPlot[Orb[p0,n,g],PlotRange->{{0,N[2Pi]},{0,N[2Pi]}},
  DisplayFunction->Identity,Frame->True,Axes->False,FrameTicks->None,
  FrameLabel->{FontForm[g,{"Helvetica",12}],"","",""}];
Movie[p0_,n_,l_]:=Table[Pict[p0,n,N[(j-1)/4]],{j,l}];
Show[GraphicsArray[Partition[Movie[{.3,.3+N[2Pi*(Sqrt[5]-1)/2]},3000,18],3]]]

For γ = 0, the map T_{0} : (q, p) ↦ (2q − p, q) can be understood completely because F (q, p) = q − p = F (T (q, p)). Each orbit is staying on a curve. For small γ, some of the invariant curves still survive. The invariant curves correspond to the caustics of the billiards. Also for the standard map, one knows explicit values, for which there exist no invariant curves any more.

Proposition 6.1.1 For |γ| > 2, the Standard map has no invariant curves winding around the torus.

Proof. According to Birkhoffs theorem, such a curve would be given by a continuous curve on which the map T would be given by a map S which has derivatives S′(q) > 0, (S^{−1})′(q) > 0 for almost all q ∈ R and some constant C. The orbit q_{n} on the curve satisfies

S(q_{n}) + S^{−1}(q_{n}) = 2q_{n} + γ · sin(q_{n}) .

The derivative of the left-hand side is positive almost everywhere. For |γ| > 2, there are intervals, where the right hand side is negative. □

Remark. Mather has refined this argument to get no invariant curves for |γ| > 4/3. MacKay and Percival have shown nonexistence of curves of |γ| > 63/64. A simple proof for nonexistence for |γ| > 1 can be found in the new book of Sinai. Numerical experiments allow to conjecture that the last invariant curve disappears at the parameter γ = 0.971635... and that this happens in a ”universal way”.

6.2 Generalized standard maps

With a smooth periodic function V ∈ C^{2}(T^{1}) one can define generalized Standard maps

T : ( \begin{matrix}q \\ p \end{matrix} ) ↦ ( \begin{matrix}2q − p + V′(q) \\ q \end{matrix} )

which leads to the recurrence equation

q_{n+1} − 2q_{n} + q_{n−1} − V′(q_{n}) = 0 .

We write this shortly as Δq = V′(q) with (Δq)_{n} = q_{n+1} − 2q_{n} + q_{n−1}. We think of Δq as a acceleration and V′ is the gradient of a potential, which is a force. The map is a discretized version of the ordinary differential equation q̈ = V (q) which becomes in the case V (q) = cos(q) the integrable pendulum equation q̈ = sin(q).

Side remark. It has recently be found out by Bobenko and Pinkall and before by Suris that there exists an integrable discretisation of the pendulum:

T_{BP} : (q, p) ↦ (2q − p + 4 · arg(1 + k · e^{−iq}), q)

with parameter |k| < 1. There exists a function F : T^{2} ↦ R

F (q, p) = 2(cos(q) + cos(p)) + k · cos(q + p) + k^{−1} · cos(q − p)

satisfying F (q, p) = F (T_{BP} (q, p)). Such a function is called an integral. The map Δ(q)_{n} = 4 · arg(1 + k · e^{−iq_{n}}) can be written with Q_{n} = e^{−iq_{n}} as a rational expression

Q_{n−1}Q_{n+1} = (\frac{Q_{n} + k}{1 + k · Q_{n}})^{2} .

Replacing the commuting coordinate functions Q_{n} by operators satisfying Q_{n}Q_{n+1} = e^{iα}Q_{n+1}Q_{n} gives a dynamical system called the quantum pendulum. The paremater α plays the role of the Plank constant. It it is 0, one gets the map T_{P Q}. Bobenko and Pinkall can explicitely describe the evolution Q_{n} ↦ Q_{n+1} if α is rational.

If the function sin(x) of the standard map is replaced by a linear function f(x) = γ|x − \frac{π}{2} (mod2π) − π| − \frac{π}{2} we get the Wojtkowsky map. Here is an orbit of the map

P=N[Pi];f[x_]:=Abs[Mod[x-P/2,2P]-P]-P/2;
T[{x_,y_},g_]:=N[Mod[{x+y+g*f[x],y+g*f[x]},2 Pi]];
Orbit[p_,n_,g_]:=Module[{},S[q_]:=T[q,g];NestList[S,p,n]];
ListPlot[Orbit[{0.2,0.3},3000,2]]

Wojtkowsky showed that for γ > 4 the Lyapunov exponent is positive almost everywhere. Furthermore, the map is conjugated to a Bernoulli shift if γ > C ∘ 4.0329. For γ = 2 cos(π/k) + 2 and k = 2, 3, … he showed that there exists a convex polygon with 2k edges such that the map has positive Lyapunov exponents outside this polygon and is periodic inside the polygon (with period k for odd k and period 2k for even k).

6.3 The generating function

Lemma 6.3.1 A generalized Standard map has the generating function

h(q, q′) = −\frac{(q′ − q)^{2}}{2} − V (q) ,

with a periodic function V ∈ C^{2}(T, R).

Proof. We calculate

p(q, q′) = h_{1}(q, q′) = q′ − q − V′(q) ,
p′(q, q′) = −h_{2}(q, q′) = q′ − q

and the equation

h_{1}(q′, q′′) + h_{2}(q, q′) = 0

reads q′′ − 2q′ + q − V′(q′) = 0. □

The transformation, which brings the first order recurrence equation q_{n−1} − 2q_{n} + q_{n+1} = V′(q_{n}) into the form

(q_{n}, p_{n}) ↦ (q_{n+1}, p_{n+1})

can be viewed as a discrete version of the Legendre transformation which allows to write a Lagrange system as a Hamiltonian system: given a sequences q_{n}, we define p_{n} = h_{1}(q_{n}, q_{n+1}) and dq_{n} = q_{n+1} − q_{n}. The Hamilton function is defined as

H(q_{n}, p_{n}) = p_{n+1}dq_{n} − h(q_{n}, q_{n+1}) = \frac{p_{n+1}^{2}}{2} + V (q_{n})

where q_{n+1}, p_{n+1} are a function of q_{n}, p_{n}. We check that the Hamilton equations

dq_{n} = H_{p_{n+1}} = p_{n+1}
dp_{n} = −H_{q_{n}} = V′(x_{n})

are satisfied. These are discrete versions of q̇ = H_{p}, ṗ = −H_{q} and are just the twist map

(q, p) ↦ (q + p + V′(q), p + V′(q)) .

Comparison between discrete Hamiltonian systems and continuous Hamiltonian systems:

Monotone twist maps

h(x, x′) generating function ∑_{j=n_{1}}^{n_{2}} h(x_{j}, x_{j+1}) action h_{1}(x_{j−1}, x_{j}) + h_{2}(x_{j}, x_{j+1}) = 0 Euler equations (second order difference eq.) h_{12} > 0 twist condition dq_{n} = q_{n+1} − q_{n} velocity p_{n} = h_{1}(q_{n}, q_{n+1}) momentum H(q_{n}, p_{n}) = p_{n+1}dq_{n} − h(q_{n}, q_{n+1}) Hamilton function (Legendre transformation) dq_{n} = H_{p_{n+1}} = p_{n+1}, dp_{n} = −H_{q_{n}} = V′(x_{n}), Hamilton equations

h(q, q′) = −\frac{(q′−q)^{2}}{2} − V (q) , discrete mechanical systems x_{n+1} − 2x_{n} + x_{n−1} = V′(x_{n}) p_{n} = q_{n+1} − q_{n} − V′(q_{n}) q_{n+1} = q_{n} + p_{n+1}, p_{n+1} = p_{n} + V′(q_{n}) Standard maps

Lagrange systems

F (x, y) Lagrange function ∫_{t_{1}}^{t_{2}} F (x, ẋ)dt action \frac{d}{dt}Fẋ = F_{x} Euler equations (second order differential equation) F_{ẋẋ} > 0 Legendre condition ẋ velocity p = F_{y}(x, ẋ) momentum H(x, p) = pẋ − F (x, ẋ) Hamilton function (Legendre transformation) q̇ = H_{p}, ṗ = −H_{q} Hamilton equations.

F (x, y) = \frac{1}{2}y^{2} − V (x) mechanical systems ẍ = V′(x) Newton equations p = F_{y} = y momentum ẋ = p, ṗ = −V′(x) Hamilton equations

6.4 Periodic orbits

Given n > 1. Define on R^{n} the functional

L_{n}(q) = ∑_{i=1}^{n} h(q_{i}, q_{i+1}) = ∑_{i=1}^{n} −\frac{(q_{i+1} − q_{i})^{2}}{2} − γ · cos(q_{i}) .

A critical point q satisfies δL_{n}(q) =) or

q_{n+1} − 2q_{n} + q_{n−1} + γ · sin(q_{n}) = 0 .

Such a critical point corresponds to a periodic orbit with period n of the Standard map.

Proposition 6.4.1 For n prime, there exists a periodic orbit of period n.

Proof. The set {q ∈ X | L(q) ≥ C} is closed and bounded and so compact. The set {L(q) > 0} is not empty. The continuous function L takes its maximum on the compact set L(q) ≥ 0 and the maximum is not on the boundary (see [40] for details). The maximum is therefore a critical point and corresponds to an orbit (q_{n}, q_{n−1}) mod2π of the twist map. □

The Hessian at a critical point is the matrix

L_{ij} = \frac{d}{dq_{i}} \frac{d}{dq_{j}} L(q_{1}, …, q_{n}) .

One calculates

L = ( \begin{matrix}b_{1} & 1 & 0 & · & · & 1 \\ 1 & b_{2} & 1 & · & · & · \\ 0 & 1 & · & · & · & · \\ · & · & · & · & 1 & 0 \\ · & · & · & 1 & b_{n−1} & 1 \\ 1 & · & · & 0 & 1 & b_{n} \end{matrix} ) ,

with b_{i} = γ · cos(q_{i}). This matrix has nonpositive eigenvalues, if the periodic orbit is a maximum.

The number of positive eigenvalues of L is called the Morse index of the periodic orbit. The Morse index is useful, because it allows to distinguish different periodic orbits.

6.5 Embedding a shift

Theorem 6.5.1 For γ large enough, there is a shift of 2 symbols embedded in the Standard map T_{γ}.

Proof. We repeat again the argument done in the Henon map. Consider the function F : R × C(X) ↦ C(X) given by

F (ϵ, q) = ϵ(q(T x) − 2q(x) + q(T^{−1}x)) + sin(q(x)) .

For ϵ = 0, every q ∈ C(X, Σ) is a solution where Σ = {0, π}. Take the special solution q(x) = x_{0}. The derivative D_{2}(0, q) = f′(q) is invertible, because cos(σ) ≠ 0 for all σ ∈ Σ. By the implicit function theorem, there exists also for small ϵ a solution q of F (ϵ, q) = 0. This q is a solution of q(T) − 2q + q(^{−1}) + γ · sin(q(x)) = 0 with γ = 1/ϵ. The map ϕ_{q}(x) = (q(x), q(T^{−1}x)) is a homeomorphism on the image K and conjugates the shift (X, T) with (K, T_{γ}). □

The argument works for a much wider class of maps. The potential cos(x) can be replaced by any 2π-periodic function f which is two times differentiable and which has at least two zeros f(σ_{i}) = 0 with f′(σ_{i}) ≠ 0. One can also consider dissipative twist maps

T_{γ,b} : ( \begin{matrix}x \\ y \end{matrix} ) ↦ ( \begin{matrix}2x − y + γf(x) \\ bx \end{matrix} )

with real parameter γ and parameter b < 1. Such maps are defined on the cylinder T × R and not on the torus unless b = 1.

The embedding of a shift in the Henon, quadratic or Standard map was a perturbative result. But one can also get explicit bounds for the γ’s which allow the embedding of a shift. We write Δ(q) = q(T) − 2q + q(T^{−1}).

Theorem 6.5.2 For γ > 4(π + 1)/3 ∼ 5.5, the Standard map has a shift with two symbols embedded.

Proof. Denote the derivative with respect to ϵ with a dot. Total differentiation of the equation

F (ϵ, q(ϵ)) = ϵ · Δq + sin(q) = 0

with respect to ϵ gives

F_{1}(ϵ, q) + F_{2}(ϵ, q)q̇ = 0

and so

q̇ = −L^{−1} · Δq

with the linear map L = F_{2}(ϵ, q) which is the linear map

Lu = ϵΔu + cos(q)u .

Because

||L̇|| = ||Δ − sin(q) · q̇|| ≤ 4(1 + π) · ||L^{−1}||

we get

\frac{d}{dϵ}||L^{−1}|| ≤ ||\frac{d}{dϵ}L^{−1}|| ≤ ||L^{−1}||^{2} · ||L̇|| ≤ 4(1 + π) · ||L^{−1}||^{3} .

This is a differential equation of the form ẏ = Cy^{3} which has the solution y(t) = (y(0)^{−2} − Ct/3)^{−2} which is finite as long as |t| < 3/C|y(0)|^{2}. We conclude that for

|ϵ| ≤ 3/4(1 + π) · ||L_{0}^{−1}|| = 3/4(1 + π)

the operator L_{ϵ} stays has a bounded inverse. As long as this stays true, one can always use the implicit function theorem to allow the embedding for a larger values of ϵ. □

6.6 The KAM theorem

KAM-theorems (Kolmogorov-Arnold-Moser) are perturbation results and deals with twist maps, which are close to integrable twist maps. It makes a statement about the smooth embedding of the dynamical system (X = T^{1}, x ↦ x + α, dx) in a Standard map (twist map). Such an embedding is possible, if |γ| is small enough and α is ”badly approximable by rational numbers” in the following sense:

An irrational number α is called Diophantine, if there exists c > 0 and μ > 2 such that

|α − \frac{p}{q}| ≥ \frac{c}{q^{μ}} , ∀p ∈ Z, ∀q > 0 .

Theorem 6.6.1 (KAM) For small enough |γ| and Diophantine α, there exist smooth periodic solutions u ∈ C^{∞}(X) such that

F (u)(x) = u(T x) − 2u(x) + u(T^{−1}x) + γ · sin(x + u(x)) = 0 .

Remark: define the function q(x) = x + u(x) as before. The embedding of the torus is then given by the smooth function ϕ : X ↦ R^{2}, x ↦ (q(x), q(T^{−1}x)).

Denote by S the rotation about the angle α/2 such that S^{2}(x) = T (x). Define the map D(u) = u(S) − u(S^{−1}). We have D^{2}(u) = u(T) − 2u + u(T^{−1}) so that F (u) = D^{2}(u) + γ · sin(x + u(x)).

A natural attempt is trying to find solutions to equations F (u) = 0 by making Newton steps

u ↦ u − \frac{F (u)}{F′(u)} .

The derivative F′(u) is then a linear map given by F′(u)v = D^{2}v + γ cos(u) · v. The naive Newton method would work if the starting point is near enough to a solution u_{0} and if F′(u_{0}) is invertible. It is the later condition which is the problem. The difficulty (and the deepness of the KAM theorem) lies in the fact that F′(u_{0}) is not invertible. The KAM theorem shows that it is still possible to find a solution. The proof of the theorem is quite technical. Our aim is only to see, where small divisor problems enter into the problem.

Lemma 6.6.2 The set of Diophantine α forms a set of measure one.

Proof. Define for m > 0 the set E_{m} as the set of α ∈ [0, 1] such that for infinitely many pairs p ∈ Z, q > m, the estimate |α − p/q| < q^{−μ} holds. The measure of E_{m} can be estimated as

|E_{m}| ≤ ∑_{q=m}^{∞} 2q^{−μ}q .

Because μ > 2, the sum ∑_{q=1}^{∞} 2q^{−μ} converges. We see that |E_{m}| → 0 for m → ∞. The complement of the Diophantine numbers α is contained in E = ⋂_{m=1}^{∞} E_{m} and has also measure 0. □

Given a smooth 1 periodic function h with Fourier development h(x) = ∑_{n∈Z} h_{n}e^{2πinx}.

Lemma 6.6.3

D^{−1}h(x) = ∑_{n∈Z,n≠0} \frac{h_{n}}{2i · sin(nπα)}e^{2πinx}

Proof.

De^{2πinx} = e^{2πinx}(e^{iπnα} + e^{−iπnα}) = e^{2πinx}2i sin(nπα) .
□

Corollary 6.6.4 If α is Diophantine and h is a periodic function h(x) = ∑_{n∈Z} h_{n}e^{2πinx}. satisfying ∑_{n} \frac{|h_{n}|}{|n|^{μ−1}} < ∞, where μ is the constant appearing in the Diophantine condition of α. Then D^{−1}h is a periodic function with convergent Fourier series.

Proof. Assume α is Diophantine then λ = e^{2πiα} satisfies for all q > 0

|λ^{q} − 1| ≥ c_{1} · q^{1−μ} .

This means

| sin(πqα)| ≥ 2c · q^{1−μ} .

A differentiable periodic function can be written as a Fourier series h(x) = ∑_{n∈Z} h_{n}e^{2πinx}. We compute

D^{−1}h = ∑_{n} \frac{h_{n}}{2i sin(nπα)}e^{2πinx} .

This sum converges if ∑_{n} \frac{|h_{n}|}{|n|^{μ−1}} < ∞. □

Because sin(πqα) can get small and is in the denominator, the boundedness of the Fourier series is a ”small denominator problem”. Such problems occur often in dynamical systems.

6.7 Aubry-Mather sets

An orbit of the twist map T on R^{2} is given by a sequence (q_{n}, q_{n−1}) ∈ R^{2}. We have seen, that the critical points of the functional L(q) = ∑_{i=1}^{Q} h(q_{i}, q_{i+1}) correspond to periodic orbits of T.

Proposition 6.7.1 For any pair (P, Q) such that P, Q have no common divisor, there exists a periodic point q_{P,Q} of the twist map T.

Proof. Given P, Q ∈ Z without common divisor. Consider the functional L on the finitedimensional space

X_{P,Q} = {{q_{i}}_{i∈Z} | q_{i} ∈ R, q_{i+Q} = q_{i} + P } .

A sequence q_{i} ∈ X_{P,Q} is called well-ordered, if

σ(x)_{k} := q_{k+u} + n ≥ q_{k} , ∀k ∈ Z ,

where u and n are uniquely determined by uP + nQ = 1. Call Y_{P,Q} the space of such well ordered sequences in X_{P,Q}, where the sequences {q_{n}}_{n∈Z} are identified with {q_{n} + 1}_{n∈Z}. One can describe any q ∈ Y_{P,Q} by q_{0} ∈ T^{1} and the condition 0 ≤ β_{k} = q_{k+u} − q_{k} + n < 1. Because ∑_{k=1}^{Q} β_{k} = 1 and β_{k} ≥ 0, the space Y_{P,Q} is compact. The continuous function L takes its maximum on the compact set L_{P,Q}. A critical point q corresponds to a periodic orbit (q_{j}, q_{j−1}) with period Q. Take the maximum of L on Y_{P,Q}. We have still to show that the maximum is not attained at a boundary (see [40]). □

The constructed periodic orbits are called Birkhoff periodic orbits.

There is a theorem of Poincaré-Birkhoff that a twist map has always 2 periodic orbits of type (P, Q). We sketch how the existence of the second periodic orbit can be proven. If q is the maximum of L, than also σ(q) is a maximum. If one connects q with σ(q) by a curve γ, then L takes its minimum somewhere on γ. Take the curve γ^{∗}, which has the hightest minimum. This is again a critical point.

If K_{n} is a sequence of compact sets. If K_{n} → K in the Hausdorff topology, then T (K_{n}) → T (K) for a Lipschitz continuous map. If K_{n} are T invariant, then also K is T invariant. Because the topological space of compact subsets of T^{2} is compact any sequence of compact T -invariant sets has accumulation points, which are compact invariant sets also.

Accumulation points (in the Hausdorff topology) of Birkhoff periodic orbits which are invariant under the map are called Aubry-Mather sets.

Theorem 6.7.2 For any α ∈ R, there exists an Aubry-Mather set K_{α}. It is in the Hausdorff topology an accumulation point of periodic orbits K_{P_{n}/Q_{n}}, where P_{n}/Q_{n} → α. K_{α} = K_{β}, if and only if α − β ∈ Z.

Aubry-Mather sets can be invariant curves or invariant Cantor sets. In the case γ = 0, every invariant curve is an Aubry Mather set. When an invariant curve ”dies” while increasing γ, it does not disappear but becomes an invariant Cantor set.

6.8 An analytic map

We will show now that the standard map family appears in an analytic map of C^{4}. This shows the analytic character of the map.

Proposition 6.8.1 The analytic map U : C^{4} → C^{4},

U : ( \begin{matrix}z \\ w \\ u \\ v \end{matrix} ) ↦ ( \begin{matrix}zwe^{z−u} \\ we^{z−u} \\ uve^{u−z} \\ ve^{u−z} \end{matrix} )

has the parameter family of two-dimensional real tori

S_{γ} = \{(z, w, u, v) ∈ C^{4} | |z| = |u| = \frac{γ}{2} , |v| = |w| = 1, z = \overline{u}, w = \overline{v}\} .

invariant, on which U is the standard map family T_{γ} : (x, y) ↦ (2 + y + γ · sin(x), y + γ · sin(x)).

Proof. This is shown by direct computation: we write on the set S_{γ}

z = \frac{γ}{2} e^{ix}, w = e^{iy}

u = \frac{γ}{2} e^{−ix}, v = e^{−iy}

Then z − u = i · γ · sin(x) and zwe^{z−u} = e^{i(x+y+γsin(x))}. □

Some properties of the map U: An integral. An integral of U is the determinant I(z, w, u, v) = det(DU) = wv of the Jacobean

DU(z, w, u, v) = ( \begin{matrix}(w + wz)e^{z−u} & ze^{z−u} & −zwe^{z−u} & 0 \\ we^{z−u} & e^{z−u} & −we^{z−u} & 0 \\ −uve^{u−z} & 0 & (v + uv)e^{u−z} & ue^{u−z} \\ −ve^{u−z} & 0 & ve^{u−z} & e^{u−z} \end{matrix} ) .

This implies that for |I| > 1 we have

det(DU^{n}) = I^{n} → ∞, n → ∞

and for |I| < 1

det(DU^{n}) = I^{n} → 0, n → ∞ .

On the invariant set I^{−1}(1), there is an other integral J = uz and the map on an invariant set wv = 1, uz = J, z ≠ 0 is given by

Ũ : ( \begin{matrix}z \\ w \end{matrix} ) ↦ ( \begin{matrix}zwe^{z−J/z} \\ we^{z−J/z} \end{matrix} ) ,

which is product of the two integrable maps

( \begin{matrix}z \\ w \end{matrix} ) ↦ ( \begin{matrix}z \\ we^{z−J/z}, \end{matrix} ) , ( \begin{matrix}z \\ w \end{matrix} ) ↦ ( \begin{matrix}zw \\ w \end{matrix} ) .

For J = \frac{γ^{2}}{4} , the map Ũ leaves invariant the torus

S̃_{γ} = \{(z, w) ∈ C^{2} | |z| = \frac{γ}{2} , |w| = 1\}

and induces there the standard map T_{γ}.

6.9 Topological entropy

Let (X, d) be a metric space and T : X ↦ X continuous. We define a sequence of metrics

d_{n}(x, y) = max_{0≤i≤n−1} d(T^{i}x, T^{i}y) .

The topological entropy h(T) is defined to be

h(T) = lim_{ϵ→0} \overline{lim_{n→∞}} \frac{log(M(n, ϵ))}{n} ,

where M(n, ϵ) is the minimal number of ϵ balls in the metric d_{n} covering the space X.

The limit lim_{ϵ→0} h_{ϵ}(T) in the definition of the number h(T) exists, because h_{ϵ}(T) grows monotonically.

We compute the topological entropy of a shift map T : X → X. The space X is equiped with the metric d(x, y) = 2^{−N}, where N is the largest non-negative integer such that x_{i} = y_{i} for all |i| < N.

Proposition 6.9.1 The shift (X, T) with X = Σ^{Z} has topological entropy log(|Σ|).

We postpone the prove to the chapter about cellular automata, where we will prove a more general statement.

6.10 Hyperbolic sets

A compact T invariant set K ⊂ T^{2} is called a hyperbolic set, if for each x ∈ K, there exists a decomposition R^{2} = E_{x}^{+} ⊕ E_{x}^{−} into one dimensional subspaces such that for two real constants C, λ < 1 and all x ∈ Λ and n ≥ 0.

||DT_{x}^{±n}v|| ≤ C · λ^{±n} · ||v||, v ∈ E_{x}^{±} .

If the whole torus T^{2} is a hyperbolic set, the map T is called Anosov. Hyperbolic sets were introduced by S.Smale in 1967 (see [34, 35, 40, 61]).

Examples:

The simplest example of a hyperbolic set is a hyperbolic fixed point or a hyperbolic periodic orbit. The decomposition of R^{2} at a fixed point x is given by the eigenspaces of the matrix dT (x). To compute the decomposition of R^{2} at a periodic point x, we compute the eigenspaces of dT^{n}(x).

The embedded shifts in the Standard map are hyperbolic sets. To show that one has to prove that there exists a splitting and that

||DT_{x}^{±n}v|| ≤ C · λ^{n}||v||, v ∈ E_{x}^{±} .

We don’t prove this here.

The cat map T (x, y) = (2x + y, x + y) is an example of an Anosov map, because the whole torus is hyperbolic. At each point, the spaces E^{±} are the eigenspaces of the matrix

( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ) .

One can take C = 1 and λ = (3 − \sqrt{5})/2.

6.11 Measuring the Lyapunov exponents

The next mathematica program allows a numerical computation of the Lyapunov exponents of the standard map.

T[{x_Real,y_Real},g_Real]:=N[Mod[{x+y+g*Sin[x],y+g*Sin[x]},2 Pi]];
A[{x_Real,y_Real},g_Real]:=N[{{1+g*Cos[x],1},{g*Cos[x],1}}];

Lya[{x_,y_},g_Real,n_Integer]:=Module[{t=0,B=IdentityMatrix[2],p={x,y}},
  Do[B=A[p,g].B;p=T[p,g]; t=t+Re[Log[B[[1,1]]]];B=B/B[[1,1]],{i,n}];t/n];
Entropy[g_]:=Module[{n=1000,m=10,R:=N[2*Pi*Random[]]},Sum[Lya[{R,R},g,n],{m}]/m];

The algorithm computes the integrated Lyapunov exponent

λ(dT_{γ}) = lim_{n→∞} ∫_{T^{2}} \frac{1}{n} log ||dT_{γ}^{n}(x, y)||dxdy ,

which we will denote shortly by Lyapunov exponent, by calculating

λ(dT_{γ}) = lim_{n→∞} \frac{1}{n} log ||dT_{γ}^{n}(x, y)||

for random points (x, y) ∈ T^{2} and averaging the values. For almost all (x, y) ∈ T^{2}, the limits λ((x, y)) exist by Fuerstenberg-Kesten’s theorem (see the billiard chapter).

Why is one interested so much in Lyapunov exponents? The first reason is the following theorem of Pesin which we give only in a special case. The proof of the theorem is technical and difficult.

Theorem 6.11.1 (Pesin) Assume a differentiable area preserving map on the torus has positive Lyapunov exponents on a set Y of positive measure. Then there exists a subset Y′ ⊂ Y of positive measure and a decomposition Y′ = ⋃_{k=1}^{n} T^{k}(Z) into disjoint pieces such that the map T^{n} : Z ↦ Z is measuretheoretically conjugated to a Bernoulli shift.

Remarks.

1) It is in general difficult to prove that one has a set of positive measure, where the dynamics is chaotic. We have proven alreay for large |γ| the existence of embedded shifts but one can show that they have their support on a set of zero measure.

2) There is a formula of Pesin which relates the metric entropy of the twist map with the Lyapunov exponents. The integral

∫_{T^{2}} λ(x, y) dx dy

is the metric entropy of the map T with respect to Lebesgue measure.

An other reason for the interest in Lyapunov exponents is that there is often a relation between the Hausdorff dimension of a closed invariant set, the Lyapunov exponent and the topological entropy of the map induced on the invariant set.

For any invariant measure on a hyperbolic set K, the Lyapunov exponent λ(x) is bigger or equal than − log(λ), where λ is the contraction rate. The Hausdorff dimension can be estimated as follows:

Theorem 6.11.2 (Fathi) Let K be a compact hyperbolic set for T. Denote with T_{K} the map T restricted to K. One has

dim_{H}(K) ≤ 2 · h(T_{K})/(− log(λ))

where λ < 1 is the constant appearing in the definition of the hyperbolic set.

Consider the compact invariant set K = ϕ(X) on which the map T induces the shift. One can show that this set is hyperbolic and that the constant λ is going to zero for γ → ∞. The topological entropy of the shift is log(2). It follows that the Hausdorff dimension of K is going to zero also as |γ| → ∞.

Let μ be a measure on a metric space (X, d). The information dimension of μ is defined as

dim_{H}(μ) = inf_{μ(S)=1} dim_{H}(S) .

There is analogue to the relation between the topological entropy and the metric entropy

h_{top}(T) = sup_{μ} h_{μ}(T) .

Theorem 6.11.3 (Young) Let T be a C^{2} diffeomorphism of a compact surface M and μ an ergodic measure with compact support. Assume the Lyapunov exponents of T are λ_{1} < 0 < λ_{2}, then

dim_{H}(μ) = h_{μ}(T) · (1/λ_{1} − 1/λ_{2}) .

This applies especially for the Standard map. If μ is the Lebesgue measure, then dim_{H}(μ) = 2 and Youngs formula is reduced to the Pesin formula h_{μ}(T) = λ_{2}.

If dim_{H}(μ) > 0, then the Hausdorff dimension of the support of μ satisfies

dim_{H}(supp(μ)) ≥ dim_{H}(μ) > 0 .

In the case of the embedded shifts, we know that the metric entropy for the Bernoulli measure is log(2) and that the Lyapunov exponents are growing like log(λ) For large enough γ, the embedded shifts live on fractals with positive Hausdorff dimension.

6.12 Search for little elliptic ilands

For a parameter value γ = 10 the standard map looks already quite stochastic. We want to zoom into a little window of the phase space in order to see if there is more structure than just dark chaos. Looking in small windows is needs more computational efforts because an orbit visits a small set only with small frequency.

T[{x_Real,y_Real},g_Real]:=N[Mod[{-2x+y+g*Sin[x],x},2 Pi]];
Orb[{x_,y_},g_Real,n_,{a_,b_,c_,d_}]:=Module[{p={x,y},s={},k=0},While[k<n,
  p=T[p,g]; If[a<p[[1]]<b && c<p[[2]]<d,Print[k,p];s=Append[s,p];k++]];s];
s=Orb[{0.1,0.1},10.,6000,{0.05,0.15,0.05,0.15}];
ListPlot[s,Frame->True,Axes->False];

6.13 Epilog

Why studiing the standard map?

• The standard map is the simplest twist map on the cylinder.

• It is the prototype of a discrete Hamiltonian system. It can be seen as the discretisation of the mathematical pendulum, or the stationnary equation of a discretised sin-Gordon equation.

• By changing the parameter there occur universal phase transitions like accumulations of period doubling or break up of tori.

• There are still many open problems like for example the question, what is the metric entropy of the map.

• Many mathematical developments can be illustrated at the standard map; KAM or Mather theory, existence of periodic orbits, ergodic theory.

As literature, we recommend [48, 40, 43].

Open research problems:

• What is the metric entropy of the standard map?

• Is the topological entropy a monotone function of γ? Does the topological entropy increase monotonically for γ → ∞?

6.14 Exercices

Topics: The standard map Fixed points of the standard map

1) Find all the fixed points of the standard map T = T_{γ}.

2) Determine the Jacobian dT (x_{i}) at these fixed points x_{i} and classify them depending on the parameter γ in elliptic, hyperbolic and parabolic fixed points.

Reversibility

A map S : X → X is called an involution, if S^{2}(x) = x for all x ∈ X. A map T is called reversible, if there exists an involution S such that

S ∘ T = T^{−1} ∘ S .

3) Show that (x, y) ↦ (−x, y + γ · sin(x)) is an involution.

4) Show that the Standard map is reversible.

5) Show that an involution maps periodic orbits into periodic orbits.

Periodic orbits

6) Compute the Morse index of the periodic orbits maximizing the functional L.

7) Show that there exists for large γ periodic orbits of the Standard map which don’t maximize the functional L.

Cellular automata

7.1 Definition of cellular automata

7.1.1 One dimensional cellular automata

Given a finite set Σ. The shift with alphabet Σ is the dynamical system σ(x)_{n} = x_{n+1} on the space X = Σ^{Z}. The set X is equiped as before with the metric

d(x, y) = 2^{− inf_{x_{i}≠y_{i}} |i|} .

For example, two sequences which coincide except at the place x_{0} ≠ y_{0} have distance 1.

A one-dimensional cellular automata is a continuous map T : X → X commuting with σ.

Especially, the shift σ and all its iterates are cellular automata.

Remark: A computer can not deal with the infinite space X. When making computer experiments, one considers the map restricted to the space of periodic sequences in Σ^{Z} which is a finite set. The map T is then just a permutation of this set. The shift, which is the prototype of a chaotic system, is then only a finite permutation. The shift cellular automaton is very boring for computers. (See Fig 4)

7.1.2 Higher dimensional cellular automata

Higher dimensional cellular automata are defined in the same way. Take the compact metric space Σ^{Z^{d}}. The metric on Z^{d} is |n − m|_{∞} = max|n_{i} − m_{i}|. Define the metric on X by

d(x, y) = 2^{− inf_{xn≠yn} |n|} .

If e_{1} = (1, 0, …, 0), e_{2} = (0, 1, …, , 0), …, e_{|Σ|} = (0, 0, …, 1) are the |Σ| basis vectors in Z^{d}, the shift σ_{i} : X ↦ X is defined as (σ_{i}x)_{n} = x_{n+e_{i}}. A cellular automata is a continuous map on X which commutes with all the shifts σ_{i}. Higher dimensional cellular automata are interesting for physical applications like for example for simulations of hydrodyamics.

Examples of cellular automata can be constructed by fixing a finite set A of indices and specifying a local function f : Σ^{|A|} → Σ which defines a cellular automata

T (x)_{i} = f({x_{i}}_{i∈A}) .

According to a theorem of Hedlund-Curtis-Lyndon, every cellular automaton arises in this fashion.

Proposition 7.1.1 (Hedlund-Curtis-Lyndon) Every cellular automata has a finite rule given by a function f : Σ^{A} → Σ.

Proof. We write Σ = {0, 1, …, |Σ|−1}. Given x ∈ X. We show that f(x) := T (x)_{0} depends only on {x_{i}, i ∈ A} where A = A(x) is some finite set with finite diameter N(x) < ∞. Assume this is not true. Then there exists a sequence of indices n_{k} ∈ Z^{d} with |n_{k}| → ∞ such that F (x + e_{n_{k}}) ≠ F (x), where (e_{n_{k}})_{n_{k}} ∈ Σ is different from 0 and (e_{n_{k}})_{n} = 0 for n ≠ n_{k}. Because x+e_{n_{k}} → x for k → ∞ the continuity of F implies that F (x+e_{n_{k}}) = F (x) eventually. This is a contradition to F (x + e_{n_{k}}) ≠ F (x). The set A(x) is invariant under the shifts σ_{i}. Assume, there exist two points x, y, where A(x) ≠ A(y). We can find a sequence of shifts σ^{(j)} ∈ Z^{d} such that σ^{(j)}(x) → y. Because f is continuous, we get f(x) = f(σ^{(j)}x) → f(y) and so f(x) = f(y). Therefore A(x) = A(y). □

The radius of a cellular automata is the minimal radius of the finite set A belonging to the local function f. A finite set A ⊂ Z^{d} has radius r if there exists a point n ∈ A such that d(n, m) ≤ r for all m ∈ A and if r is the smallest number with this property. A ball with radius r is called a cube with radius r.

Proposition 7.1.2 Every cellular automata is isomorphic to a cellular automata with radius 1. Every cellular automata is isomorphic to a cellular automata with |Σ| = 2.

Proof. Assume a cellular automata has radius r. Make a partition of Z^{d} into cubes of radius r each containing N = (2r + 1)^{d} lattice points. The midpoints of these cubes form again a lattice isomorphic to Z^{d} and if we take a new index set J with |Σ|^{N} points, each configuration x ∈ X = Σ^{Z^{d}} can be mapped bijectively into a configuaration y ∈ Y = J^{Z^{d}}. The map T on X is mapped into an isomorphic cellular automaton S on Y. The inverse of this transformation allows to take the index set Σ having only two elements in expense of increasing the radius of the automaton. □

Remark. Cellular automata can also be defined for other graphs G different from Z^{d}. If each lattice point looks the same, which is for example the case, if G is the Cayley graph of a group which has finitely many generators, then the above results hold also.

Proposition 7.1.3 There are Σ^{Σ·(2r+1)^{d}} different cellular automata of radius r in dimension d with alphabet Σ.

Proof. In Z^{d}, a ball B_{r} of radius r has (2r + 1)^{d} elements. The cellular automata are in one to one correspondence with the maps

Σ × B_{r} → Σ .

There are |Σ|^{|Σ×B_{r}|} such functions. □

We see for example that in 2 dimensions, there are 2^{18} = 262144 cellular automata with radius 1 and alphabet Σ = {0, 1}. ”Life” is one of them.

7.2 Topological entropy of cellular automata

We have defined the topological entropy for a continuous map T of a compact metric space (X, d) as

h(T) = lim_{ϵ→0} \overline{lim_{n→∞}} \frac{log(M(n, ϵ))}{n} ,

where M(n, ϵ) is the minimal number of ϵ balls in the metric

d_{n}(x, y) = max_{0≤i≤n−1} d(T^{i}x, T^{i}y) .

covering the space X.

We restrict us in the following to one-dimensional cellular automata T. A space-time diagram of T is an infinite array

N × Z → Σ, (n, k) ↦ (T^{n}x)_{k} .

Define R(N, K) to be the number of distinct rectanges of width K and height N which occur in a space-time diagram of T.

Proposition 7.2.1 The topological entropy of a one dimensional cellular automaton is given by

h(T) = lim_{K→∞} lim_{N→∞} \frac{log(R(N, K))}{N} .

Proof. Given two (N, 2K + 1)-rectangles A, B in the space-time diagram. Enumerate the rows of A and B starting from the bottom with A_{1}, A_{2}, …, A_{N} and B_{1}, …, B_{N} and take two elements x, y ∈ X such that

(T^{j}(x_{−K}), … T^{j}(x_{−1}), T^{j}(x_{0}), T^{j}(x_{1}), …, T^{j}(x_{K})) = A_{j},

and

(T^{j}(y_{−K}), … T^{j}(y_{−1}), T^{j}(y_{0}), T^{j}(y_{1}), …, T^{j}(y_{K})) = B_{j} .

Because A_{j} = B_{j} if and only if d(T^{j}(x), T^{j}(y)) < 2^{−K} we get that

A = B ⇔ d_{N}(x, y) < 2^{−K}.

If we have two points x, y ∈ X with d_{N}(x, y) ≥ 2^{−K}, then we have two different rectangles.

a) M(N, 42^{−k}) ≤ R(N, 2K + 1)

b) M(N, 2^{−k}/4) ≥ R(N, 2K + 1) Proof of a). Take for each of the R(N, 2K + 1) rectrangles A a point x such that

(x_{−K}, … x_{−1}, x_{0}, x_{1}, …, x_{K}) = A_{1} .

This gives a finite set Y ⊂ X with R(N, 2K + 1) points. Every point x ∈ X has distance ≤ 22^{−K} to one of the points in Y. The R(N, 2K + 1) balls of radius 42^{−k} with midpoints in Y cover X. This proves a). Proof of b). Two different points in Y have distance ≥ 2^{−K}/2.

We need therefore at least R(N, 2K + 1) balls of radius 2^{−K}/4 to cover X. The statements a) and b) together give

R(N, 2(K + 4) + 1)) ≥ M(N, 2^{−(K+2)}) ≥ R(N, 2K + 1)

and

lim_{N→∞} \frac{log(R(N, 2(K + 4) + 1))}{N} ≤ lim_{N→∞} \frac{log(M(N, 2^{−(K+2)}))}{N} ≤ lim_{N→∞} \frac{log(R(N, 2K + 1))}{N} .

For K → ∞, the left and right limits converge to the same number. The limit in the middle gives the topological entropy. □

Corollary 7.2.2 The topological entropy of a cellular automata is finite.

Proof. R(N, K) ≤ |Σ|^{K+2r·N} so that h(T) ≤ r · log(|Σ|). □

Corollary 7.2.3 The topological entropy of the shift with alphabet Σ is log(|Σ|).

Proof. T = σ. R(N, K) = |Σ|^{K+N}.

h(σ) = lim_{K→∞} lim_{N→∞} log(|Σ|)\frac{K + N}{N} = log(|Σ|) .
□

The topological entropy of a higher dimensional cellular automata is in general infinite. The topological entropy of a shift in one direction is infinite. It follows (see exercice) that the topological entropy of the game of life is infinite. Ergodic theory shows that every measure theoretical dynamical system with finite metric entropy can be found inside the game of life.

Without proof, we mention the following quite recent theorem

Theorem 7.2.4 (Hurd,Kari and Culik) For any ϵ > 0, there exists no algorithm which will take a cellular automaton T and decide whether h(T) > ϵ. For any ϵ > 0. there is no algorithm which will take a cellular automaton T and return h̃ such that |h(T) − h̃| < ϵ.

We do not give a proof. The strategy of the proof is to relate the problem of calculating the entropy to the ”Stoppoing problem of Turing machines, which is a undecidable problem: there exists no algorithm which takes a Turing machine and decides whether it halts or not. Turing machines themselves form interesting dynamical systems.

7.3 Examples of cellular automata

Examples of one-dimensional automata:

• The shift σ and any of its iterates are cellular automatons.

k=2; Init[n_]:=Table[Floor[k*Random[]],{n}]; T[x_]:=Mod[RotateLeft[x],k];
R=NestList[T,Init[100],100];ListDensityPlot[R,Mesh->False];

• Take Σ = {0, …, k − 1} and impose on Σ the group operation modulo k. Inspired from the Henon map, we take

T (x)_{i} = x_{i+1} − 2x_{i} + x_{i−1} + x_{i}^{2} − c modk

with a parameter c ∈ Σ. Here is an example with k = 7, c = 3. See Fig 1. Computer experiments are done by restricting the map to periodic sequences. Here we have taken period 100 and 100 time steps.

k=7; Init[n_]:=Table[Floor[k*Random[]],{n}];
T[a_]:=Mod[RotateLeft[a]-2a+RotateRight[a]+a^2+3,k];
ListDensityPlot[NestList[T,Init[100],100],Mesh->False,ColorOutput->RGBColor];

• An elementary cellular automaton is a one dimensional automaton, where Σ = {0, 1} and T (x)_{i} depends only on x_{i−1}, x_{i}, x_{i+1}. Here is an example of such a cellular automaton. (See Fig. 2).

k=2; Init[n_]:=Table[Floor[k*Random[]],{n}];
T[a_]:=Mod[RotateLeft[a]+RotateRight[a],k];
ListDensityPlot[NestList[T,Init[100],100],Mesh->False,ColorOutput->RGBColor];

Such a cellular automat is defined by a function T (x)_{i} = g(x_{i−1}, x_{i}, x_{i+1}). There are 256 = 2^{(2^{3})} such functions and therefore 256 elementary cellular automatas. Wolfram attached to each of these automaton a number

∑_{i,j,k∈{0,1}} g(i, j, k)2^{4i+2j+k}

and calls it the rule of this automaton. In our example, the function was g(i, j, k) = i + k ( mod 2). The mathematica calculation

g[i_,j_,k_]:=Mod[i+k,2]; Sum[g[i,j,k]*2^(4*i+2*j+k),{i,0,1},{j,0,1},{k,0,1}]

shows that we were looking at rule 90. If the function g is replaced by P (g)(i, j, k) = g(k, j, i), we get an isomorphic cellular automaton, and the parity transformation P (x)_{n} = x_{−n} is the isomorphism. Also, the replacement C(g)(i, j, k) = 1−g(1−i, 1−j, 1−k) gives an isomorphic automaton and the transformation P (x)_{n} = 1 − x_{n} (exchanging 0 and 1) is the isomorphism between the two systems. Many of the 256 different rules give isomorphic systems. Counting the equivalence classes reduces the number 256 to 88.

• The random number generator of Wolfram This is a elementary cellular automata with I = {0, 1} and rule f(x)_{n} = x_{n−1} + x_{n+1} + x_{n} + x_{n}x_{n+1}. It has the rule

g[i_,j_,k_]:=Mod[i+j+k+j*k,2];Sum[g[i,j,k]*2^(4*i+2*j+k),{i,0,1},{j,0,1},{k,0,1}]

= 30. Such an automata is used in Mathematica as a pseudo random generator.

• Elementary reversible cellular automata are very similar to elementary cellular automata but they have the property that they are invertible. Take the alphabet Σ = {(0, 0), (0, 1), (1, 0), (1, 1)} and write every element z ∈ X = Σ^{Z} as z = (x, y) with x, y ∈ {0, 1}. Take again a function g : Σ^{3} → Σ and define

T (x, y)_{i} = (y_{i} + g(x_{i−1}, x_{i}, x_{i+1}), x_{i}) (mod2) .

Such automata are invertible and have the inverse

T^{−1}(x, y)_{i} = (y_{i}, x_{i} + g(y_{i−1}, y_{i}, y_{i+1}) (mod2) .

It suffices to see the dynamics of x coordinate because y[t] = x[t − 1].

k=2;Init[n_]:={Table[If[j==5||j==6,0,1],{j,n}],Table[If[j==5||j==6,0,1],{j,n}]};
T[{x_,y_}]:={Mod[y+RotateLeft[x]+RotateRight[x],k],x};
R=NestList[T,Init[100],100];ListDensityPlot[Table[R[[j,1]],{j,100}],Mesh->False];

The next automaton (rule 250) is isomorphic to the Bobenko automaton.

k=2;Init[n_]:={Table[Floor[k*Random[]],{n}],Table[Floor[k*Random[]],{n}]};
k=2;Init[n_]:={Table[If[j==5||j==5,0,1],{j,n}],Table[If[j==70,0,1],{j,n}]};
T[{x_,y_}]:={Mod[y+1+RotateLeft[x]*RotateRight[x],k],x};
R=NestList[T,Init[100],100];
ListDensityPlot[Table[R[[j,1]],{j,100}],Mesh->False];

(* three dimensional version
k=2;n=10;R0:=Table[If[i==5 && j==6,0,1],{i,n},{j,n}];
m[o_]:=Mod[o-1,n]+1;
T[{x_,y_,z_}]:={Mod[Table[y[[i,j]]
  +x[[m[i],m[j+1]]]*x[[m[i],m[j-1]]]+ x[[m[i],m[j+1]]]*x[[m[i-1],m[j]]]
  +x[[m[i+1],m[j]]]*x[[m[i-1],m[j]]]+ x[[m[i+1],m[j]]]*x[[m[i],m[j-1]]]
  +z[[m[i],m[j+1]]]*z[[m[i],m[j-1]]]+ z[[m[i],m[j+1]]]*z[[m[i-1],m[j]]]
  +z[[m[i+1],m[j]]]*z[[m[i-1],m[j]]]+ z[[m[i+1],m[j]]]*z[[m[i],m[j-1]]]
  ,{i,n},{j,n}],k],z,x}
R={R0,R0,R0}; Do[ListDensityPlot[R[[1]],Mesh->False];R=T[R],{10}];

• Randomized elementary cellular automata. Take again the alphabet Σ = {(0, 0), (0, 1), (1, 0), (1, 1)} and write every element z ∈ X = Σ^{Z} as z = (x, y) with x, y ∈ {0, 1}. Take again a function g : Σ^{3} → Σ and define

T (x, y)_{i} = (x_{i+1}, x_{i} + g(y_{i−1}, y_{i}, y_{i+1}) (mod 2) .

If we hide the first coordinate of a point x ∈ X, we see only the element y ∈ {0, 1}^{Z}. The sequence of x_{i} can be viewed as a random sequence which adds noise to the cellular automaton with alphabet {0, 1}. (see Fig 3. )

k=2; Init[n_]:=Table[Floor[k*Random[]],{n}];
T[a_]:=Mod[Init[Length[a]]+RotateLeft[a]+RotateRight[a],k];
ListDensityPlot[NestList[T,Init[100],100],Mesh->False,ColorOutput->RGBColor];

• Shift register sequences These are special one-dimensional celluar automata of the form

F (x)_{n} = f(x_{n−1}, …, x_{n−r}) .

The shift itself can be defined as a shift register sequence which is an elementary cellular automaton. Shift register sequences can be understood by studiing at the map f from I^{r} → I.

Examples of higher dimensional cellular automata:

• The soliton model This is a one-dimensional ”integrable” cellular automata found by Bobenko and Pinkal. The particles are scattering like solitons.

• Game of life The game of life is a two dimensional cellular automata which has been invented in 1970 by John Horton Conway. Given a configuation x ∈ X, we define a map Z^{2} → N by S_{x}(n) = ∑_{|m−n|−1} x_{m} which is the number of ”living” neighbors of the cell n. The terminology of the game is that a cell n ∈ Z^{2} is ”living” if x_{n} = 1 and ”dead”, if x_{n} = 0. Define

T (x)_{n} = x_{n}1_{S_{x}(n)∈{2,3}} + (1 − x_{n})1_{S_{x}(n)=3} .

This means that a living cell will keep beeing alive if and only if it has 2 or 3 neighbors. If there are more or less neighbors, it will die from overcrowdedness ir loneliness. A dead cell will come alive if and only if it has 3 living neighbors.

• The Hexagonal Lattice Model (HLG) Designed by Frisch,Hasslacher and Pomeau in 1985. The rules are designed to conserve particle number and momentum at each vertex. The model is not deterministic because a random number generator is used. One can look at it as a skew product of two cellular automata, the first one (X, T_{1}) is the shift generating the random sequences, the second one (X, T_{2}(y)) is the three dimensional cellular automaton depending on a parameter y. The map is given by the skew product T (x, y) = (T_{1}(x), T_{2}(x, y)).

7.4 A map on subshifts

We will take present Hedlunds point of view of subshifts, namely that cellular automata are factor maps on a class of topological dynamical systems.

A topological dynamical system is a pair (X, T), where X is a compact metric space and T is a Z^{d} action of homeomorphism of X.

A topological dynamical system is expansive, if there exists ϵ > 0 such that for all x ≠ y ∈ X, there exists n ∈ Z^{d} such that d(T^{n}x, T^{n}y) ≥ ϵ. It is zero dimensional if X is zero dimensional which means that there exists a basis containing clopen sets.

A subshift (X, T) is a topological dynamical system, where X is a compact subset of A^{Z^{d}}, where A is a finite set with the discrete topology and T is the Z^{d} action given by the shifts on X.

Theorem 7.4.1 (Reddy) A zero dimensional, expansive dynamical system is isomorphic to a subshift.

Proof. Construct a finite topological generator by taking a partition of X into n clopen sets X_{i}, where each of the sets has diameter ≤ ϵ. Each point defines an orbit and so an encoding into A^{Z}, where A labels the partition. The expansiveness assures that the encoding is injective. □

Remark. The converse is not true since a subshift needs not to be expansive (take a periodic subshift). However, an aperiodic subshift is expansive.

A factor map (X, T) → (Y, S) is a continuous surjective map ϕ : X → Y such that ϕT = Sϕ. It is an isomorphism if it is injective.

A cellular automaton defines a factor map on the space of subshifts. Definition. A block map is a factor map ϕ : X → Y, where X, Y are subshifts such that ϕ(x)_{n} depends only on finitely many {x_{k}}_{k∈n+F}. We have seen that the Curtis-Hedlund-Lyndon theorem assumres that any cellular automaton is a block map.

7.5 Properties of subshifts invariant under CA maps

In this section we see that cellular automata are natural since they are the endomorphisms of the categorie of topological dynamical systems, namely subshifts As a consequence, they preserve properties which are inherent for subshifts.

First a remark about categories. The set of subshifts is a subset of morphisms of the categorie of zero dimensional topological spaces, namely the expansive morphisms. It is typical in the theory of dynamical systems that one iterates a map on the space of categories: consider a class of morphisms which is a categorie and consider morphisms on this categorie. The next iteration in our case would be to consider maps on the space of cellular automata maps but we do not go so far.

Properties which are invariant under cellular automata:

• Topological transitive: exists a dense orbit.

• Almost periodic = minimal: every orbit is dense.

• Uniquely ergodic: exists exactly one compact invariant measure.

• Strictly ergodic: minimal and uniquely ergodic.

• Uniquely ergodic and discrete spectrum: the Koopman operator with respect to the invariant measure has discrete spectrum.

• Dense set of periodic orbits: x periodic orbit:

• ∃n ∈!!Z^{d}, T^{n}x = x.

• Prime: Every factor of (X, T) is either trivial or isomorphic. Example: Double the ones in a Sturmean sequence.

• Totally minimal: No factor is a finite permutation.

• Completly positive entropy: all non trivial factors have positive directional entropy.

• Zero directional topological entropy:

• Topologically strongly mixing: U, V open. Exists n ∈!!Z such that U ∩ T^{n}V ≠ ∅.

• topologically weakly mixing: X × X is topologically transitive.

• uniquely ergodic, strong mixing: μ(U ∩ T^{n}V) → μ(U) · μ(V).

• uniquely ergodic, weakly mixing subshifts: X × X is ergodic.

• sophic: factors of a subshift of finite type.

• Bernoulli: Ornstein has shown that factors of Bernoulli shifts are Berno ulli shifts. Properties of subshifts which are not invariant under cellular automata:

• Chaotic in the sense of Devaney: 1d: topological transitive and dense set of periodic orbits and not finite. General: additional sensitive dependence on initial conditions. There exists δ > 0 such that for all x ∈ X, ϵ > 0, there exists y ∈ X, d(x, y) < ϵ and n ≥ 1 with d(T^{n}x, T^{n}y) ≥ δ.

• Expansive: Periodic factors are not expansive.

7.6 Minimal subshifts

We will see in this section that the set of minimal subshifts forms a compact metric space in the Hausdorff topology of all subshifts. Almost periodic cellular automata are special continuous maps on this space.

We have seen that the set of compact subsets of A^{Z^{d}} is a compact metric space with the Hausdoff topology

d(X, Y) = inf_{ϵ>0}{ X ⊂ Y_{ϵ}, Y ⊂ X_{ϵ}} ,

where X_{ϵ} = {x ∈ A^{Z^{d}} | d(x, X) ≤ ϵ} is the ϵ-collar of X.

Proposition 7.6.1 The set of minimal subshifts X_{minimal} is compact in X.

Proof. We have to show that it is closed. Given X_{n} → X and x_{n} ∈ X_{n} converging to x ∈ X. Since the orbit closures X(x_{n}) converge to X, we have X(x) ⊂ X. Assume there exists z ∈ X \ X(x). There exists z_{n} → z with z_{n} ∈ X_{n}. Since every accumulation point of X_{n} = X(z_{n}) contains z and so X(z), we must also have z ∈ X. This shows that X(x) = X and X is minimal. □

Since minimal subshifts are the closure of the set of periodic subshifts, they are called almost periodic. In order to investigate a cellular automaton on subclasses of minimal subshifts, one has to refine the topology in order to get a compact metric space.

Remark. It is known that the topological entropy is upper semi-continuous on the compact set of subshifts: if X_{n} → X then h_{top}(X) ≥ lim sup h_{top}(X_{n}). In general there is no continuity. A case, where one has continuity of h_{top} if X_{n+1} ⊂ X_{n}).

Without proof, we mention:

Theorem 7.6.2 (Sigmund) Assume d = 1. Subshifts with zero entropy are generic in the set of all subshifts.

We consider now constructions of almost periodic subshifts.

Substitutions. A substitution is a map S : A ↦ A^{∗}; it will be extended to a map A^{∗} ↦ A^{∗} and A^{N} ↦ A^{N} by concatenation, that is, S(a_{1}a_{2} ⋯ a_{n}) = S(a_{1})S(a_{2}) ⋯ S(a_{n}). A substitution S is called primitive if there exists a k such that for all a ∈ A the word S^{k}(a) contains at least one copy of every symbol. Without loss of generality, one can assume that there exists a ∈ A such that S(a) starts with a. Iterating S on a then gives a fixed point z^{+} ∈ A^{N}. Let z be any element in A^{Z} satisfying z_{n} = z_{n}^{+} for n ∈ N. The substitution dynamical system (X_{S}, T) is now defined by

X_{S} = {x ∈ A^{Z} | x = lim_{j→∞} T^{n_{j}}z and n_{j} → ∞} .

It is possible to generate the sequences in X_{S} directly from the substitution. The system (X_{S}, T) is strictly ergodic.

Quasiperiodic subshifts (e.g. Sturmean sequences). Consider irrational rotations θ ↦ θ + α_{i} on the circle T^{1} = R/Z. For a countable union of half-open intervals J = [0, β) ⊂ T^{1} and a point θ_{0} in T^{1}, we consider the sequence z = z(θ_{0}) given by

z_{n} = 1_{J}(n · α + θ_{0}) ∈ {0, 1}^{Z} .

We denote by X_{J} the compact subset of A^{Z} obtained by taking the closure of the orbit of z(θ_{0}). It is independent of θ_{0} and the system (X_{J}, T) is strictly ergodic.

Dyadic group construction e.g. Toeplitz systems. The group G of dyadic integers is given as a set as in G = {0, 1}^{N}. The addition is obtained by writing the sequence as a formal power series

g = ∑_{n=1}^{∞} g_{n}2^{n}, h = ∑_{n=1}^{∞} h_{n}2^{n}

and adding these. The zero element is (0, 0, …). Call 1 = (1, 0, 0, …) the one element and let U_{i} : G → G, g ↦ g + e_{i} be ergodic group translations. Define G_{n} = {g ∈ G | g_{1} = g_{2} = … = g_{n−1} = 0, g_{n} = 1}. Given a sequence v ∈ {0, 1}^{N} define J = ⋃_{v_{n}=1} G_{n}. In the case d = 1 and where U is translation by one, the sequence z_{n} = 1_{J}(U^{n}(0)) is called a Toeplitz sequence.

General group construction. if G is any compact topological group and U_{i} are ergodic group translations and R is a U_{i}-invariant ring of sets J which are a disjoint countable union of sets J_{i} = U_{i} − V_{i}, where U_{i}, V_{i} are open sets with positive Haar measure. Denote with λ the Haar measure.

Proposition 7.6.3 Given J ∈ R The subshift defined by z_{n} = 1_{J}(U^{n}(g)) is independent of g, strictly ergodic and every translation has zero entropy and pure point spectrum.

Proof. Minimality follows, if we show unique ergodicity and that every open nonempty set has positive measure. To show unique ergodicity, one has to show that every finite configuration occurs with prescribed positive frequency, if it occurs.

To show that we use the fact that for every J ∈ R and ϵ > 0, we can find continuous functions f_{1}, f_{2} such that f_{1} ≤ 1_{J} ≤ f_{2} with ∫ f_{1} − f_{2}dλ ≤ ϵ. Let F_{n} ⊂ Z^{d} be a van Hoove sequence. We have

∫ f_{1} dλ ≤ |F_{n}|^{−1} ∑_{m∈F_{n}} 1_{J}(U^{m}g) ≤ ∫ f_{2} dλ

and the limit exists uniformly.

Given a translation U, the spectrum of X_{J} is a subgroup of the spectrum of U since ϕ(g) = 1_{J}(U^{n}g) is a measure theorietical factor. □

Remark. In general, these subshifts are measure theoretically isomorphic to the group translation. It can however not be topological since the subshifts are expansive contrary to the group translation.

Cellular automata maps. More strictly ergodic subshifts can be obtained from given ones by looking at the images under cellular automata maps. Often, one reaches like this systems with pure point spectrum. As an example, consider the Thue-Morse system X defined by a substitution a− > ab, b− > ba. If we take the cellular automaton ϕ with rule 3 (defined by 110− > 1, 111− > 1 all others − > 0), then ϕ(X) is a Toeplitz system. The relation between Thue-Morse and Toeplitz is due to Dekking (see [?]).

7.7 Cellular automata on circle subshifts

Circle subshifts are simple, need few datas to be described completely and are invariant under cellular automata maps.

Denote by J the ring generated by half-open intervals [a, b) ⊂ T, with the convention [a, a) = ∅. Every J ∈ J is a finite union of half-open intervals. Given α = (α_{1}, …, α_{d}) and θ ∈ T^{1}, let x_{n} = 1_{J}(θ + n · α) ∈ {0, 1}^{Z^{d}}. Assume that the α_{i} are rationally independent. Then the orbit closure of x is strictly ergodic, and independent of θ We call it a circle subshift.

Circle subshifts of {0, …, N − 1}^{Z^{d}} are defined analogously as the orbit closure of x_{n} = f(θ + n · α), where f is a function taking values in {0, …, N − 1} on half-open intervals that partition T.

Lemma 7.7.1

Cellular automata leave circle subshifts invariant.

Proof. The proof is constructive. □

Remark. In one dimension, the spectrum of a circle subshift is {e^{2πiknα}} where k is the symmetry in J.

7.8 Turing machines as dynamical systems

A Turing machine is a dynamical system given as follows. Let Y = X × S = {0, 1}^{Z} × S, where S is a finite set consisting of states. The set S contains 0, which is called the halte state. The set {(…, 0, …)} × S is called the empty tape. The set X is the space of tapes. Tapes X_{f} which contain only finitely many 1 are called data. Given three maps

f : {0, 1} × S → {0, 1}
g : {0, 1} × S → S
h : {0, 1} × S → {−1, 0, 1} ,

we define the continuous map on the compact metric space Y

T (x, s) = (σ^{h(x_{0},s)}(…, x_{−2}, x_{−1}, f(x_{0}, s), x_{1}, x_{2} …), g(x_{0}, s)) .

This dynamical system is called a Turing machine. The description of a Turing machine needs only finitely many information, because the three involved functions map finite sets into finite sets. The set X × {0} is called the halte set. One step of a Turing machine can be described as follows: the Turing machine with tape x and state s replaces the tape at the place 0 with the entry f(x, s), moves the tape h(x, s) steps and goes into the state s.

Known facts about Turing machines: Every computation which can be done by any of the known computations can be done by Turing machines. The question of what actually can be computed is probably beyond the scope of mathematics. There is a widly accepted dogma, the Church thesis (1934) which says that everything which can be computed can be computed with a partial recursive function. It is proven that such functions can be computed with Turing machines.

The set of pairs (T, x) where T is a Turing machine and x ∈ X_{f} is an input data, is countable. We can encode therefore the set of such pairs in X_{f}. Denote with Z ⊂ X_{f} the set of all so obtained pairs (T, x). A subset Z of X_{f} is called decidable, if there exists a Turing machine, which tells after finitely many steps, whether a given x ∈ X_{f} is in Z or not.

Lemma 7.8.1 (Turing) The subset H ⊂ T M of all pairs (T, x) of Turing machines T, and datas x which halt with data x is not decidable.

Proof. Assume this set is decidable. Then there exists a Turing machine HALT which gives from the input (T, x) the output HALT(T, x) = true, if T halts with the input x and HALT(T, x) = false else. We construct now a Turing machine DIAGONAL, which has as an input an input x and does the following

 Read[x];
 Stop:=Halt[(x,x)];
 While Stop==True Do Stop:=True.
 Print[Stop].

We want now to decide, if (DIAGONAL, DIAGONAL) is in the set H or not. If DIAGONAL ∈ H, then the variable Stop was T rue, which means that the program DIAGONAL is running for ever and so

Halt[(DIAGONAL, DIAGONAL)] = False

and therefore DIAGONAL ∉ H. Assume now DIAGONAL ∉ H, then the variable Stop was F als, which means that

Halt[(DIAGONAL, DIAGONAL)] = true

which implies DIAGONAL ∈ H. This is also a contradiction and our assumption of the existence of Turing machine HALT was wrong. □

Remark. There exists a so called universal Turing machine, which can simulate all Turing machines. The universal Turing machine takes a Turing machine with input (T, x) as input and gives out the output, which the machine x would produce.

An interesting problem is the problem of the ”busy beaver”. The problem is to construct a Turing machine which has n states not counting the halt state 0. There are two symbols on the tape. The machine starts on the empty tape and should write as many marks on the tape as possible. For n = 1, 2, 3, 4, the optimal solutions are known. For n = 5, Heiner Marxen has built a Turing machine in 1989 which produces for n = 5 4098 marks. The orbit is has length 11’798’826. Here is our Mathematica program which simulates this Turing machine

F[{state_,symbol_,position_}]:=Module[{newsym,move,newstate},
If[state==1 && symbol==b, newsym=m ; move=-1; newstate=2];
If[state==1 && symbol==m, newsym=m ; move=-1; newstate=1];
If[state==2 && symbol==b, newsym=m ; move= 1; newstate=3];
If[state==2 && symbol==m, newsym=m ; move= 1; newstate=2];
If[state==3 && symbol==b, newsym=m ; move=-1; newstate=1];
If[state==3 && symbol==m, newsym=m ; move= 1; newstate=4];
If[state==4 && symbol==b, newsym=m ; move=-1; newstate=1];
If[state==4 && symbol==m, newsym=m ; move= 1; newstate=5];
If[state==5 && symbol==b, newsym=m ; move= 1; newstate=0];
If[state==5 && symbol==m, newsym=b ; move= 1; newstate=3];
{newstate,newsym,position+move}];
T[{state_,band_,position_}]:=Module[{newstate,newsym,newpos,newband=band},
{newstate,newsym,newpos}=F[{state,band[[position]],position}];
newband[[position]]=newsym;
If[newpos>Length[band],newband=Append[newband,b]];
If[newpos<1,newband=Prepend[newband,b];newpos++];
{newstate,newband,newpos}];
Orbit[{state_,band_,position_}]:=Module[{z={state,band,position},k=0},
While[Not[state==0],z=T[z];k++;If[Mod[k,10]==0,Print[k]]];Print[z[[2]]]];
Orbit[{1,{b},1}]

7.9 Dynamical systems with higher dimensional time

Dynamical systems with higher dimensional time G appear more and more often in literature about dynamical systems.

We will consider two-dimensional time G = Z^{2}, G = N × Z or G = N^{2}. Given two commuting maps T_{1}, T_{2} on a set X. We define for each n = (n_{1}, n_{2}) ∈ N^{2} the transformation T^{n}(x) = T_{1}^{n_{1}}T_{2}^{n_{2}}. This gives a N^{2} action. If T_{2} is invertible, we have a N × Z action, if both T_{1}, T_{2} are invertible, we have a Z^{2} action.

Examples.

• Let T_{1} be a one-dimensional cellular automaton map on X = Σ^{Z} and T_{2} be the shift. This is a N × Z action. If T_{1} is invertible, we have a Z^{2} action.

• Given X = Σ^{Z^{2}}. Denote with T_{1} the shift x_{n} ↦ x_{n+(1,0)} and with T_{2} the shift x_{n} ↦ x_{n+(0,1)}. This is a Z^{2} action. If X = Σ^{N×Z}, the first shift T_{1} is no more invertible and we have a N × Z action.

Given a (not necessarily invertible) topological dynamical system (X, T) where X is a compact set and T is a continuous map on X. We say, such a dynamical system is space-time chaotic, if there exists a continuous invertible map S : X → X commuting with T such that the N × Z action is conjugated to a N × Z shift on a compact subset of X. For a invertible topological dynamical system, we require to have a Z^{2} shift embedded.

7.10 Higher dimensional time systems in other fields

We list some examples, where higher dimensional dynamical systems in other fields:

Statistical mechanics Parts of the thermodynamic formalism for Statistical mechanics on a lattice Z^{d} can be extended to statistical mechanics of a Z^{d} action on a compact metric space. This theory has been developed by Bowen Ruelle and others. One can see it as a step to push the mathematics of equilibrium statistical mechanics into more deterministic domains.

Percolation problems Given any Z^{d} dynamical system (X, T_{1}, … T_{d}, m) and a measurable set Y ⊂ X. The points on the orbit

O(x) = {T^{n}x = T_{1}^{n_{1}}T_{2}^{n_{2}} … T_{d}^{n_{d}}x | n ∈ Z^{d}}

which belong to Y are called activated sites on the lattice O(x). Connected components of activated points are called clusters. Choosing a one-parameter family of sets Y_{p} with m(Y_{p}) = p and Y_{p} ⊂ Y_{q} (p < q), one finds a critical point p_{c}, above which there is an infinite cluster and below which, there are only finite clusters. See for examples of this dependent percolation. Usually, percolation problems deal with a lattice (for example Z^{d}) and the rule that each site is activated independently with probability p.

Combinatorial number theory The multiple Birkhoff recurrence theorem for a Z^{d} action (X, T_{1}, …, T_{d}) on a compact metric space X states that there exists a point x ∈ X and a sequence n_{i} → ∞ such that T^{n_{i}}x → x for all i = 1, … d. Fürstenberg showed, how this result can be used to prove Van der Waerdens theorem which tells that any partition of N into two disjoint sets A ∪ B has the property, that one of the two sets contains arithmetic progressions of arbitrary length. He obtained also various generalizations of this combinatorial result by ergodic theoretical tools or methods from topological dynamics.

7.11 Open research problems

• For cellular automata with almost periodic initial conditions it would be interesting to characterize attractors of CA maps.

• For Turing machines, there is the ”Busy beaver” problem. Given a Turing machine with |S| = n states. What is the longest orbit starting in the ”empty tape” set, landing in the halte set among all Turing machines sith n states? This is maybe a hopeless question to answer even for say n = 10.

7.12 Epilog

Why studiing cellular automata?

• Cellular automata are very convenient for computer programming. They are completely discrete systems and need not to be discretised like partial differential equations or ordinary differential equations.

• They can simulate real physical processes like hydrodynamics.

• They are mathematically interesting in connection with questions of complexity.

• They can be used as random number generators (like for example used in Mathematica).

A bit of history: Cellular automata seem having been invented by von Neumann and S.Ulam in the fifties. First mathematical research was initiated by Hedlund in 1969 [29]. The game of life, a special cellular automata was the first program running on the connection machine. Wolfram suggested in 1984 to use cellular automata as a practical approximation metheod for simulating physical systems. See his reprint book [60]. The connection machine with around 65’000 processors and several gigabytes of main memory is the most intersting machine existing for investigations of lattice gas. Alternatively, in Los Alamos, mainly Crays together with Sun workstations are used. See the story ”discrete fluids” by B. Hasslacher in the book ”From Cardinals to Chaos” [28]. The game of life was invented in 1970 by J.Convay. Cellular automata with almost periodic initial condition were introduced in [31].

Coupled map lattices

Coupled map lattices are discrete parabolic partial differential equations. Examples are discrete nonlinear Schrödinger equations. The linearisations are discrete Schrödinger equations.

Take an infinite array of copies of a dynamical system f : M → M where M is a compact subset of R^{n} or a torus R^{n}/Z^{n}. We build a new dynamical system on M^{Z} by defining

T (q)_{n} = f(q_{n}) + ϵ(f(q_{n+1}) − 2f(q_{n}) + f(q_{n−1})) .

This system is called a coupled map lattice. For ϵ = 0 the array of dynamical systems is running uncoupled. The parameter ϵ is determining the diffusion rate. Usually one takes a one parameter family of dynamical systems f_{γ}, where the parameter γ allows to move into stable or unstable dynamics.

In slang language: we study local chaos with global diffusion. The hope is to get parameter values, where one has ”space time (spatial temporal) chaos.

a=3.66; eps=0.3; M=20; K=200; Init=Table[Random[],{M}];
f[x_]:=a*x*(1-x);g[x_]:=eps*(RotateLeft[x]+RotateRight[x]-2x)+x;h[x_]:=g[f[x]];
Orb=Module[{y=Init,s={}},Do[y=h[y],{1000}];Do[s=Append[s,y];y=h[y],{K}];s];
Pha=Module[{s={}},Do[s=Append[s,Point[{j,Orb[[i,j]]}]],{j,M},{i,K}];s];
Show[Graphics[Pha]];

A coupled map lattice has the property that it is a continuous map T : M^{Z} → M^{Z}, which commutes with the shift σ(q)_{n} = q_{n+1}. What is the relation of a coupled map lattice with a cellular automaton? If M is a finite group, f : M ↦ M is a continuous map on this group and and ϵ ∈ Z, then the coupled map lattice T is a special cellular automaton. A coupled map lattice can be viewed as a continuum limit, where the discrete set Σ is replaced with the continuum M. Note also that if a computer calculates a coupled map lattice, he works on finite sets and computes therefore the evolution of a cellular automaton.

8.1 Embedding a two dimensional shift in coupled map lattices

We write Δ(q)_{n} = q_{n+1} − 2q_{n} + q_{n−1} for q ∈ M^{Z} A coupled map lattice

(T x)_{n} = ϵ · Δf(q)_{n} + f(q_{n})

is an infinite chain of coupled dynamical systems which is in the limit ϵ = 0 a lattice of uncoupled non-interacting maps. Because the time-evolution T commutes with space translation (Ux)_{n} = x_{n+1}, one can consider the system as a dynamical system with time Z × N.

Example 1. We consider first an example of a coupled map lattice, where M = T^{1} is a one-dimensional circle and f is the map

f_{γ}(q) = q + α + γ · sin(q) .

This dynamical system is called the standard circle map. Let Y = M^{Z} be the infinite torus. The coupled map lattice is the dynamical system (Y, T), where

T (q)_{n} = f_{γ}(q_{n}) + ϵ · Δf_{γ}(q)_{n} (mod2π) .

The space Y with the product topology is compact and T is continuous. The pair T, U generate a dynamical system with time N × Z. We show that this coupled map lattice (X, T, U) has a shift X = {0, π}^{N×Z} embedded in the sense that there exists a compact T and U invariant set K of X such that the dynamics of (K, T, U) is isomorphic to the shift (X, σ_{1}, σ_{2}): there exists a homeomorphism ϕ : X → K such that

ϕ ∘ σ_{1} = T ∘ ϕ, ϕ ∘ σ_{2} = U ∘ ϕ .

Theorem 8.1.1 For γ large enough and ϵ small enough, the coupled map lattice (T, U) has a two-dimensional shift embedded.

Proof. Define for δ ∈ R the map g_{δ}(q) = δ · (q + α) + sin(q). We define a map F : R^{2} × C(X) → C(X) by

F (ϵ, δ, q) = δ · q(T) − g_{δ}(q) − ϵ · δΔg_{δ}(q) .

If F (ϵ, δ, q) = 0, then ϕ : X → Y

ϕ(x) = {q(σ_{1}^{n}x)}_{n∈Z} ∈ Y = M^{Z}

conjugates the system (X, σ_{1}, σ_{2}) with (K, T, U), where K = ϕ(X). The fact ϕ ∘ σ_{1}(x) = U ∘ ϕ(x) is clear. The conjugation ϕ ∘ σ_{2}(x) = T ∘ ϕ(x) follows from F (ϵ, δ, q) = 0. If ϕ : X → K is bijective, then the two N × Z systems are isomorphic.

The construction of q for small δ and ϵ is done again with the implicit function theorem. For q(x) = x_{0}, we get

F (0, 0, q) = sin(q(x)) = 0

and F_{3}(0, 0, q)u = cos(q)u is invertible. We can find with the implicit function theorem for ϵ and δ small enough some solution q. □

Example 2). We can take the quadratic map f : C ↦ C, z ↦ x^{2} + c. For large enough |c|, and small enough ϵ, the coupled map lattice T has a two dimensional shift embedded.

For ϵ = 0, the image of the map ϕ is a copy of infinitely many Julia sets.

Example 3). As a third example, we can also take an invertible system (M, f) as the local system. Take M = T^{2} and the standard map f_{γ} : M → M. As above, we define the coupled map lattice

T (x)_{n} = f_{γ}(x_{n}) + ϵ(f_{γ}(x_{n+1}) − 2f_{γ}(x_{n}) + f_{γ}(x_{n−1})) (mod2π) .

The same proof shows again that for γ large enough and ϵ small enough, the coupled map lattice T is space-time chaotic.

Example 4) The idea of a coupled map lattice can be iterated. As soon we have a oneparameter family of maps which a shift embedded, we can build with this map f, a coupled map lattice. Especially, we can take a coupled map lattice (X, T) as a local dynamical system (M, f) and use it to construct a new coupled map lattice. If the new coupling is small enough there is a N^{3} Bernoulli shift embedded. This procedure can be iterated and we obtain coupled map lattices of higher and higher dimension.

Example 5) We have defined reversible elementary cellular automata. This idea how to make a system invertible can be applied also to the coupled map lattice case. Given the standard map family f_{γ} : M → M on M = T^{2}. Define X = (M × M)^{Z} and the map

T (x, y)_{i} = (f_{γ}(y_{i}) + ϵ · Δf_{γ}(x)_{i}, f_{γ}(x_{i})) .

For large |γ| and small |ϵ|, we have a map with an invertible two-dimensional shift on {0, π}^{Z^{2}} embedded. Also this construction can be iterated analoguous to example 4.

With the following Mathematica program, we compute an (γ, ϵ) array of pictures. Each picture shows the orbit T^{100}(q), … T^{120}(q) of a coupled map lattice with 20 periodically coupled one-dimensional standard circle maps (see fig 1.)

f[a_,r_,z_]:=N[Mod[z+a+r*Sin[z],N[2Pi]]];
g[e_,x_]:=N[Mod[e*(RotateLeft[x]+RotateRight[x]-2x)+x,N[2Pi]]];
P[M_,K_,e_,a_,r_]:=Module[{y=Table[Random[],{M}],t={}},
  h[x_]:=g[e,f[a,r,x]]; Do[y=h[y],{100}];Do[t=Append[t,y];y=h[y],{K}];t];
S[M_,K_,e_,a_,r_]:=Show[Graphics[Module[{s={},Q=P[M,K,e,a,r]},
  Do[s=Append[s,Point[{j,Q[[i,j]]}]],{j,M},{i,K}];s]],
  Frame->True,Axes->False,AxesLabel->None,FrameTicks->None];
Show[GraphicsArray[Table[S[20,20,k*0.01,0.23,r],{k,0,5},{r,0,5}]]]

Fig 2. shows an orbit of three coupled standard maps.(See fig 2.)

e=0.5;g=1.0;K=3;M=1000; R:=2*{Random[],Random[]};
f[{q_,p_}]:=N[Mod[{2q-p+g*Sin[q],q},2 Pi]];
l[x_]:=N[Mod[e*(RotateLeft[x]+RotateRight[x]-2x)+x,N[2Pi]]];
h[x_]:=l[Table[f[x[[i]]],{i,3}]];
Q=Module[{y=Table[R,{K}],t={}},Do[t=Append[t,y];y=h[y],{M}];t];
Show[GraphicsArray[Table[ListPlot[Table[Q[[j,k]],{j,M}],Frame->True],{k,K}]]]

8.2 Discrete neural networks

Some neural networks can be considered as generalisations of cellular automata or coupled map lattices depending on whether the internal states of the neurons are discrete or not.

We could call such networks discrete time neural networks.

Take a graph Γ (which can be infinite) and a dynamical system (M, f) where M ⊂ R^{n}. Take a new dynamical system X = M^{Γ} and define

T (x)_{n} = f(x_{n}) + ϵ · ∑_{d(m,n)=1} f(x_{m}) − f(x_{n}) .

The points of Γ are the neurons. The sites of the graph connect the neurons.

More realistic neural networks are obtained by taking different dynamical systems (M, f_{m}) located at the edges m of the graph. The map would be

T (x)_{n} = f(x_{n}) + ϵ · ∑_{d(m,n)=1} f_{m}(x_{m}) − f_{n}(x_{n}) .

In the human brain, there are about 10^{12} neurons and one distinguishes today several hundreds of different types of neurons. Because neurons are also interacting in different way. We can generalise our model by considering for each neuron a different coupling given by a map g : M × M → M. The dynamical system (M^{Γ}, T) with

T (x)_{n} = f_{n}(x_{n}) + ϵ · ∑_{d(m,n)=1} g_{n}(f_{m}(x_{m}), f_{n}(x_{n}))

is modelling our neural network.

Assume each dynamical system (M, f_{n}) allows the embedding of a shift given by an equation F_{n}(x) = 0 (for example, all the f_{n} are circle maps all the f_{n} are twist maps or all the f_{n} are coupled map lattices) and assume there is a uniform bound C such that ||Dg_{n}||, ||Df_{n}||, ||DF_{n}^{−1}|| ≤ C for all n. Then (M^{γ}, T) allows the embedding of a shift for ϵ small enough.

8.3 Overview

Denote with M a manifold, with G a graph, with Σ a finite set.

System  Time Space State Equation
Ordinary differential equation R  {0}  R^{n}, M ẋ = f(x)
Maps  Z  {0}  R^{n}, M x^{(m+1)} = f(x^{(m)})
Cellular automata  Z  Z^{d}, G Σ  x_{n}^{(m+1)} = f({x_{k}^{(m)}}_{|k−n|=1})
Coupled map lattices  Z  Z^{d}, G R^{n}, M x_{n}^{(m+1)} = f({x_{k}^{(m)}}_{|k−n|=1})
Partial differential equations R  R^{d}  R^{n}  ẋ = f(x, {∂_{y_{i}}x})
Neural networks  R  Z^{d}, G R^{n}, M ẋ = f({x_{k}^{(m)}}_{|k−n|=1})

8.4 Open research problems

The coupled map lattice construction can be iterated. This is interesting, when the coupling is nonliear. We get so systems with higher and higher complexity. A natural question is to understand the dynamics of the coupled map construction and to characterize an attractor in the class of systems, where it is defined.

8.5 Epilog

Since coupled map lattice (CML) theory was introduced by Kaneko in the early 1980s [37], the interest for such spatial extended systems has expanded into many fields. Originally CML’s were introduced as models for studying spatio-temporal chaos. Today, CMLs are used as models for many biological, chemical, physical and engineering problems (see e.g. [38, 39] and references therein).

Why studiing coupled map lattices?

These maps are exciting from different point of view. First of all they are mathematical models located between finite dimensional dynamical systems, cellular automata and models in statistical mechanics. They can be viewed as discretized partial differential equations and can toy models for turbulence. A reason for its popularity is the simplicity of the definition. The coupled map lattice construction can be iterated [36].

The physical motivation for the CML construction, comes from the need to model a variety of complicated behavior observed in experiments. The most challenging one is the problem of turbulence or spatio-temporal chaos that is seen for example in the Bénard convection. Until recently, there was no deterministic model of an extended system, where one had succeeded in proving that spatio-temporal chaos exists. However, Bunimovich and Sinai proved in [10] that for a CML consisting of one-dimensional map with strong statistical properties and with weak space interaction, spatio-temporal chaos exists. Also for weakly coupled hyperbolic maps, this is shown to be true by Gundlach and Rand in [25, 26, 27].

8.6 Exercices

Topic: Two dimensional cellular automata

”Speed of light” in a cellular automaton.

1) Given a 2-dimensional cellular automaton over the alphabet Σ = {0, 1, … k} having the property that T (0^{Z^{d}}) = 0^{Z^{d}}. Start with the initial condition given by x_{n} = 1 if and only if n = (0, 0). At time k ∈ N, a finite region of the space Z^{2} will be occupied by 1’s, the rest will consist of 0’s. Estimate the diameter of the occupied region {n ∈ Z^{2} | T^{k}(x)_{n} = 1} from above in terms of the radius of the automaton. Find the best bound by constructing a cellular automaton, which reaches this bound.

Topological entropy of two-dimensional cellular automata

2) A space-time diagram of a two dimensional cellular automaton is an infinite array

(n, k_{1}, k_{2}) ↦ (T^{n}x)_{(k_{1},k_{2})}, x ∈ Σ^{Z^{2}} .

Let R(N, K_{1}, K_{2}) be the number of distinct cubes of length K_{1} and width K_{2} and hight N appearing in a space time diagram of T. Show that the topological entropy of T is

h(T) = lim_{K_{1},K_{2}→∞} lim_{N→∞} \frac{log(R(N, K_{1}, K_{2}))}{N} .

3) Show that the topological entropy of a two dimensional cellular automaton can be infinite. Prove especially that ”Life” has infinite topological entropy.

4) Find a cellular automaton of radius 1 and |σ| = 2^{n} which satisfies T^{n} = σ, where σ is the shift. What is the topological entropy of this automaton?

Differential equations

9.1 The Lorenz system

The famous dynamical system studiied first by E. Lorenz is the differential equation

\begin{aligned}ẋ & = & −10x + 10y \\ ẏ & = & 28x − y − xz \\ ż & = & −8/3z + xy \end{aligned}

The original derivation of these equations can be described as follows. A two-dimensional fluid cell is warmed from below and cooled from above and the resulting convective motion is modelled by a partial differential equation. The variables are expanded into an infinite number of modes and all except three of them are put to zero. The variable x is proportional to the intensity of convective motion, y is proportional to the temperature difference between ascending and descending currents and z is proportional to the distortion from linearity of the vertical temperature profile.

A numerical integration with Runge-Kutta integration of the Lorenz system is done by the following program. The output can be seen in Fig. 0.

RungeKuttaStep[f_,y_,y0_,dt_]:=Module[{k1,k2,k3,k4},
  k1=dt N[f /. Thread[y->y0]     ];k2=dt N[f /. Thread[y->y0+k1/2]];
  k3=dt N[f /. Thread[y->y0+k2/2]];k4=dt N[f /. Thread[y->y0+k3  ]];

  y0+(k1+2*k2+2*k3+k4)/6];
Orbit[f_,y_,y0_,t_,d_]:=Module[{q=y0,s={}},
  Do[q=RungeKuttaStep[f,y,q,d];s=Append[s,q],{Round[t/d]}];s]
Orbit[{10(y-x),-x z+ 28x-y,x y-8/3 z},{x,y,z},{0,1,0},40.,0.01];
Show[Graphics3D[{Line[%]}],Axes->Automatic]

It is in general very difficult to study a specific system per se. It is better to parametrize the system which allows to look at bifurcations and generic properties of the system. For the Lorenz system one considers the parametrized Lorenz equations

\begin{aligned}ẋ & = & σ(y − x) \\ ẏ & = & rx − y − xz \\ ż & = & xy − bz . \end{aligned}

The parameters σ > 1, r > 0, b > 0 have a physical interpretation. σ is the Prandl number (quotient of viscosity and thermal conductivity), r depends amoung other things on the temperature difference of the heated layer, b is a constant depending on the geometry of the fluid cell.

9.2 An existence theorem for differential equations

We will prove here the Cauchy-Picard existence theorem for differential equations in R^{n}. Because there is no difference in the proof when replacing R^{n} by a general Banach space, we do it in this generality. Note also that we used already differential equations in Banach spaces in the proof of the explicit bound for which the standard map has a shift embedded.

Let X be a Banach space and I an interval in R. Let f be a mapping I × X → X . A differentiable mapping u : J → X of an open ball J ⊂ I in X is called solution of the differential equation

ẋ = f(t, x)

if we have for all t ∈ J the relation

u̇(t) = f(t, u(t)) .

In the proof of the existence theorem we will use a result from the Henon chapter:

Lemma 9.2.1 Let X = \overline{B_{b}(x_{0})} ⊂ X and assume ϕ is a differentiable map X → X. If for some λ < 1 and for all x ∈ X, ||ϕ(x) − ϕ(y)|| ≤ |λ| · ||x − y|| and

||ϕ(x_{0}) − x_{0}|| ≤ (1 − λ) · b

then ϕ has exactly one fixed point in X.

Theorem 9.2.2 (Cauchy-Picard Existence theorem) Let f : I × X → X be continuous in the first coordinate and locally Lipschitz continuous in the second. Then the following holds: for every (t_{0}, x_{0}) ∈ I × X, there exists an open interval J ⊂ I with midpoint t_{0}, such that on J, there exists exactly one solution of the differential equation ẋ = f(t, x).

Proof. There exists an interval J(t_{0}, a) = (t_{0} − a, t_{0} + a) ⊂ I and a ball B(x_{0}, b), such that

M = sup{||f(t, x)|| | (t, x) ∈ J(t_{0}, a) × B(x_{0}, b)}

as well as

k = sup\{\frac{||f(t, x_{1}) − f(t, x_{2})||}{||x_{1} − x_{2}||} | (t, x_{1}), (t, x_{2}) ∈ J(t_{0}, a) × B(x_{0}, b), x_{1} ≠ x_{2}\}

are finite. Define for r < a the Banach space

X_{r} = C(\overline{J}(t_{0}, r), X) = \{y : \overline{J}(t_{0}, r) → X , y continuous\}

with norm

||y|| = sup_{t∈\overline{J}(t_{0},r)} ||y(t)||

Let V_{r,b} be the open ball in X_{r} with radius b around the constant mapping t ↦ x_{0}. For every y ∈ V_{r,b} we define

ϕ(y) : t ↦ x_{0} + ∫_{t_{0}}^{t} f(s, y(s))ds

which is again an element in X_{r}. We prove now, that for r small enough, ϕ is a contraction. A fixed point of ϕ is then a solution of the differential equation ẋ = f(t, x), which exists on J = J_{r}(t_{0}). For two points y_{1}, y_{2} ∈ V_{r}, we have by assumption

||f(s, y_{1}(s)) − f(s, y_{2}(s))|| ≤ k · ||y_{1}(s) − y_{2}(s)|| ≤ k · ||y_{1} − y_{2}||

for every s ∈ \overline{J}_{r}. Thus, we have

||ϕ(y_{1})−ϕ(y_{2})|| = || ∫_{t_{0}}^{t} f(s, y_{1}(s))−f(s, y_{2}(s)) ds|| ≤ ∫_{t_{0}}^{t} ||f(s, y_{1}(s))−f(s, y_{2}(s))|| ds ≤ kr||y_{1}−y_{2}|| .

On the other hand, we have for every s ∈ \overline{J}_{r}

||f(s, y(s))|| ≤ M

and so

||ϕ(x_{0}) − x_{0}|| = || ∫_{t_{0}}^{t} f(s, x_{0}(s)) ds|| ≤ ∫_{t_{0}}^{t} ||f(s, x_{0}(s))|| ds ≤ M · r .

We can apply the above lemma, if kr < 1 and Mr < b(1 − kr). This is the case, if r < b/(M + kb). □

Examples: The Lipschitz continuity of the function f is necessary. The differential equation

ẋ = f(x) = x^{1/3}

has not a unique solution with initial condition x(0) = 0, because x(t) = 0, x(t) = (±\frac{2}{3}t)^{3/2}.

Local existence does not imply that the solution exists for all times. The differential equation in R

ẋ = x^{2}

has a solution x(t) = 1/(1/x(0) − t) which reaches infinity after a finite time if x(0) ≠ 0.

Corollary 9.2.3 Either a solution x(t) exists for all times t > 0, or there exists t_{∞} > 0 such that for t → t_{∞} the solution satisfies ||x(t)|| → ∞.

Proof. Assume, we have a solution in the interval [0, t_{0}) By Zorn’s lemma, there exists a maximal time t_{∞} ∈ R^{+} ∪ {∞} such that x(t) is defined in [0, t_{∞}). Assume t_{∞} < ∞ and ||x(t_{∞})|| < ∞. It follows from the local existence theorem that there exists an interval [t_{∞}, t_{∞} + ϵ) such that the solution can be continued to [0, t_{0} + ϵ). This contradicts the maximality of t_{∞}. □

Remarks. The above prove of the existence theorem can be modified in different ways. One can allow complex time z. The solutions are then analytic functions x(z). One can also show that if f is r times differentiable, then (x_{0}, t) ↦ x(t) is r times differentiable.

9.3 Global existence of the Lorenz flow

A subset Y of R^{n} which has the property that if x(t) ∈ Y then for all s > t also x(s) ∈ Y is called a trapping region.

Proposition 9.3.1 (Lorenz) There exists a bounded ellipsoid E ⊂ R^{3} which is a trapping region for the Lorenz flow. The time-one map T maps E into the interiour of E.

Proof. The function

V = rx^{2} + σy^{2} + σ(z − 2r)^{2}

satisfies

V̇ = −2σ(rx^{2} + y^{2} + bz^{2} − 2brz) .

Define D = {V̇ ≥ 0}. This is a bounded region. Let c the maximum of V in D. Define E = {V ≤ c + ϵ} for some small positive ϵ. This is a region containing D. If a point x lies outside E then it also lies outside D and so V̇ ≤ 0. In fact V̇ ≤ −δ for some positive δ depending on ϵ. A point x starting outside E has a value of V such that V is decreasing and within a finite time, the trajectory will enter the ellipsoid D. Furthermore, all trajectroies pass inwards through the boundary of E. A trajectory, once within E will remain there forever. □

Remark. A function, which is nondecreasing is also called a Lyapunov function.

9.4 The Poincaré Bendixon theorem

If ẋ = f(x) be a differential equation. Assume x_{0} is an initial condition for which the solution x(t) is defined for all t ∈ [0, ∞). The set O^{+}(x_{0}) = {x(t) t > 0} is called the semi-orbit of x_{0}. Define the omega limit set as

ω(x_{0}) = ⋂_{s≥0} \overline{O^{+}(x(s))} .

Theorem 9.4.1 (Poincaré-Bendixen) Let ẋ = f(x) be a differential equation defined in the plane R^{2}. Assume x_{0} is an initial point such that x(t) stays in a bounded region. If ω(x_{0}) contains no point y with f(y) = 0, then ω(x_{0}) is a single periodic orbit to which x(t) converges.

Proof. An smooth arc s : [0, 1] ↦ γ(s) ∈ R^{2} is called transverse to the vectorfield f if at every point x ∈ γ, the vector f(x) and a tangent vector of γ are linearily independent at x. Claim: If a solution x(t) crosses γ more than once, the successive crossing points form a monotonic sequence in γ. Denote with x(t_{1}) = γ(s_{1}), x(t_{2}) = γ(s_{2}), the first two crossing times. We assume that s_{2} ≥ s_{1} (reparametrize else γ by s′ = 1 − s if s_{1} < s_{2}). The union of the two smooth arcs {x(t) | t_{1} ≤ t ≤ t_{2}} and {γ(s) | s_{1} ≤ s ≤ s_{2}} is a closed Jordan curve. Jordan’s theorem states that such a curve divides the plane into two different regions. For t > t_{2}, the solution x(t) stays in one of these regions. For the next crossing x(t_{3}) = γ(s_{3}) one has therefore s_{3} ≥ s_{2}. It follows that no more than one point of any transverse arc γ can belong to the set ω(x_{0}). Given y_{0} ∈ ω^{+}(x_{0}). Because x(t) stays by assumption in a bounded region, the solution x(t) is defined for all times and stays in ω^{+}(x_{0}). Since there exists no stationary point in ω^{+}(x_{0}), we can find a transverse arc γ passing through y_{0}. Because ω^{+}(x_{0}) ∩ γ can have only one intersection and x(t) returns arbitrary close to y_{0}, the orbit through y_{0} is a single periodic orbit. □

Poincaré-Bendixon’s theorem tells us that differential equations in the plane do not a chaotic dynamics. The only attractors which can appear are periodic orbits attracting stationary solutions. Two stationary solutions can be connected by curves.

We must go therefore to at least three dimensions in order to get interesting dynamics. The Lorenz system is an example.

9.5 Equilibrium points

A steady state or equilibrium point of a differential equation ẋ = f(x) is a zero of f.

Example. For the Lorenz system the equilibrium points are readily calculated by solving the three nonlinear equations. One gets the equilibrium point O = (0, 0, 0) and for r > 1 also other equilibrium points C_{±} = (±b\sqrt{r − 1}, ±b\sqrt{r − 1}, r − 1). These are the ”holes” in the attractor. There is a natural symmetry (x, y, z) ↦ (−x, −y, z) which maps the Lorenz system into itself. The two points C_{±} are mapped into each other and one has only to study one point, in order to know everything about the other.

Given an equilibrium of a differential equation ẋ = f(x) and an equilibrium point x_{0}. The point x_{0} is called stable or Lyapunov stable, , if for any neighborhood U of x_{0}, there exists a neighborhood V of x_{0} such that a flow starting in V is staying for all positive times in U. The point x_{0} is called asymptotically stable or a ”sink” if there exists a neighborhood V of x_{0} such that every orbit starting in x is converging to x_{0} in positive time. The first concept of stability is important in conservative dynamical systems and it is subtle and usually hard to verify. The second strong kind of stability can be decided by looking at the linearized system u̇ = df(x)u around the point x_{0}.

Lemma 9.5.1 The linear system u̇ = Au has 0 has an asymptotically stable fixed point if and only if all the eigenvalues α_{i} of A satisfy Reα_{i} < 0.

Proof. u(t) = e^{At} is the solution of u̇ = Au. There exists a unitary matrix U which brings A into Jordan normal form U^{∗}AU = BDiag(α_{j}) + N, where N^{n} = 0. We can write

e^{Bt} = U^{∗}e^{At}U = Diag(e^{α_{j}t}) ∑_{j=0}^{n−1} \frac{t^{j}}{j!} N^{j} .

||e^{At}|| = ||e^{Bt}|| ≤ e^{max Re(α_{j})}C(1 + |t|^{n−1}) .

The other direction is shown as follows. Assume Re(α_{j}) ≥ 0. Take an eigenvector ψ. Then ||e^{At}ψ|| = |e^{α_{j}t}| · |ψ|. □

Two continuous maps T_{i} : U_{i} ⊂ R^{n} ↦ R^{n}, are called topologically conjugated on the open sets U_{i} ⊂ R^{n}, if there exists a homeomorphism ϕ : R^{n} ↦ R^{n} such that T_{2}(x) = ϕT_{1}ϕ^{−1}(x) for all x ∈ U_{2}. Two differential equations ẋ = f_{i}(x) are called topologically conjugated on some open sets U_{i} if their time t maps T_{i} : U_{i} ↦ R^{n} are topologically conjugated for all t in a neighborhood of 0.

Theorem 9.5.2 (Grobmann-Hartmann)

a) Assume x_{0} is a fixed point of a differentiable map T : R^{n} ↦ R^{n}. Assume A = dT (x_{0}) is hyperbolic, which means that the eigenvalues of A are not on the unit circle. Then, T is topologically conjugated to A in some neighborhood of x_{0}.

b) Given a stationary point x_{0} of a differential equation ẋ = f(x). Assume B = df has eigenvalues α_{i} and Re(α_{i}) ≠ 0. Then the system is topologically conjugated to ẋ = Ax in some neighborhood U of x_{0}.

Proof. a) Reduction to a global problem. Take a smooth function ϕ_{ϵ} which satisfies ϕ_{ϵ}(x) = 1 for x ∉ B_{2ϵ}(x_{0}) and ϕ_{ϵ}(x) = 0 for x ∈ B_{ϵ}(x_{0}). The map S = T + ϕ_{ϵ} · (A − T) is equal to T for |x − x_{0}| < ϵ and equal to A for |x − x_{0}| > 2ϵ. If can write now S(x) = Ax + f(x), where

||f′||_{∞} → 0

for ϵ → 0. We have to show that S : R^{n} ↦ R^{n} is conjugated by a map H(x) = x + h(x) to the linear map A : R^{n} ↦ R^{n} if S = A + f and ||f′||_{∞} is small enough. The conjugating equation and its linearization. The condition H ∘ A(x) = T ∘ H(x) is

h(A(x)) − Ah(x) = f(x + h(x)) .

We have to solve this equation for h. Look first at the linearized problem

(Lh)(x) := h(A(x)) − Ah(x) = f(x) .

Solving the linearized problem. The matrix A is hyperbolic. Write R^{n} = E_{+} ⊕E_{−}, where E^{+} is spanned by the eigenvectors of A belonging to the eigenvalues |λ_{i}| < 1 and E^{−} is spanned by the eigenvectors to A belonging to the eigenvalues |λ_{i}| > 1. We can decompose the problem to find a solution by solving

h_{±}(A(x)) − Ah_{±}(x) = f_{±}(x) ,

where h = h_{+} +h_{−}, f = f_{+} +f_{−} is the decomposition satisfying f_{±}, h_{±} ∈ E^{±}. The operator U : C(R^{n}) ↦ C(R^{n}), h ↦ h(A) as well as its inverse have norm 1. Because

||(U − A_{+})^{−1}|| = ||U^{−1} ∑_{n=0}^{∞} A_{+}^{n}U^{−n}|| ≤ \frac{1}{1 − λ}

||(U − A_{−})^{−1}|| = || − A_{−}^{−1} ∑_{n=0}^{∞} A_{−}^{−n}U^{n}|| ≤ \frac{λ}{1 − λ} < \frac{λ}{1 − λ}

with λ = max{||A_{+}||, ||A_{−}^{−1}||} < 1, we can solve for h by

h = h_{+} + h_{−} = (U − A_{+})^{−1}f_{+} + (U − A_{−})^{−1}f_{−} .

Solving the nonlinear problem. Define Φ(h)(x) = f(x + h(x)) − f(x). We need to solve the equation

Lh = Φh + f

in C(X). This can be done by

h = (1 − L^{−1}Φ)^{−1}L^{−1}f

if 1 − L^{−1}Φ is invertible. This is possible, if L^{−1}Φ is a contraction. The later is indeed the case if ||f′||_{∞} is small enough:

||L^{−1}Φh_{1} − L^{−1}Φh_{2}|| ≤ \frac{1}{1 − λ} · ||Φh_{1} − Φh_{2}||_{∞} ≤ \frac{1}{1 − λ} · ||f′||_{∞} · ||h_{1} − h_{2}|| .

b) Taking the time-t map

T : x(0) ↦ x(t) ,

of the differential equation ẋ = f(x) and the time t map

A : x ↦ e^{tB}x

of the differential equation ẋ = Bx. For all t > 0, the matrix A is hyperbolic because Reμ_{i} ≠ 0 for all eigenvalues μ_{i} of the matrix B. The maps T and A are locally conjugated by a). Therefore the flows are topological conjugated in a neighbohood of the stationary solution x_{0}. □

Remark. An other proof goes as follows. We have to find a zero of

F (h, f) = (U − A)h(x) − f(x + h(x)) = 0 .

This can be done with the implicit function theorem: For f = 0, h = 0 is a solution. The derivative

DF (h, f)(x)u(x) = (U − A)u(x) − f′(x + h(x))u(x)

is invertible, if ||f′||_{∞} is small enough. There exists therefore for small ||f||_{∞} also a solution of F (h, f) = 0.

Corollary 9.5.3 Given a stationary point x_{0} of a differential equation ẋ = f(x). Assume A = df has eigenvalues α_{i} and Re(α_{i}) < 0. Then the stationary point x_{0} is asymptotically stable.

Proof. The notion of asymptotic stability is invariant under topological conjugation. Since Re(α_{i}) ≠ 0, we can conjugate the system ẋ = f(x) near x_{0} to the linear system u̇ = Au, which is by the above lemma asymptotically stable, also ẋ = f(x) is asymptotically stable. □

We apply this now to the Lorenz system, which has the linearisation

( u̇ ; v̇ ; ẇ ) = ( \begin{matrix}−σ & σ & 0 \\ r − z & −1 & 0 \\ y & x & −b \end{matrix} ) ( \begin{matrix}u \\ v \\ w \end{matrix} ) .

The stability can be checked with mathematica.

Eigenvalues[{{-s,s,0},{r-z,-1,0},{y,x,-b}}]

which gives

\{−b, \frac{−1 − s + \sqrt{1 − 2 s + 4 r s + s^{2} − 4 s z}}{2} , \frac{−1 − s − \sqrt{1 − 2 s + 4 r s + s^{2} − 4 s z}}{2} \} .

There is the following picture of stability of the three equilibrium points for σ = 10, b = 8/3. for r ∈ (0, 1), the origin is globally stable. For r > 1, the origin is unstable, there is a one-dimensional unstable and a two-dimensional stable direction. For 1 < r < r^{∗}24.7368 = (σ(σ + b + 3)/(σ − b − 1), the two points C_{i} are asymptotically stable. For For r > r^{∗} the two stationary points are unstable. An interesting point is the parametervalue r^{∗}. When increasing r through r^{∗}, the eigenvalues cross the imaginary axes. The stationary points C_{i} get unstable and one sees that there is a stable periodic orbit expelled nearby. Such a phenomena is called a Hopf bifurcation.

9.6 Periodic orbits and knots

A periodic orbit of a differential equation ẋ(t) = f(t, x(t)) is a solution x(t) satifying x(t) = x(t + T) for all t ∈ R. The number T ∈ (0, ∞) is called the period of the orbit. If T = 0, then we have equilibrium points. We don’t count such points as periodic orbits.

For the Lorenz system, the z-axis x = y = 0 is invariant and all trajectories which start on the z-axis remain on it and tend towards the origin O. All trajectories which rotate around the z− axis do so in a clockwise direction when viewed from above to the plane z = 0. This follows from the fact that if x = 0 then ẋ > 0 and if y = 0 then ẋ < 0. A partial description of periodic orbits in the systems can be given by counting the number of times they wind around the z-axes.

A closed curve in R^{3} is called a knot. Given a differential equation ẋ = f(t, x) in R^{3}, every periodic orbit is by definition a knot. Especially, the Lorenz system has knots as periodic orbits.

9.7 Poincaré sections

Given a differential equation ẋ = f(x) defining a flow in R^{3}. Given a two dimensional plane Σ in R^{3}. There exists an open subset U ⊂ Σ, such that for x = x ∈ U f(x) is not tangential to Σ and such that the orbit x(t) is hitting Σ again transversely in a point x(t_{1}) at a first time. This defines a map T : U ⊂ Σ → Σ which is called a Poincaré map of the flow.

Examples:

1) Let x(t) = x(t + T) be a periodic orbit. Let Σ be a plane normal to f(x(0)). There exists an open neighborhood U of x(0) ∈ Σ where the first return map is defined.

2) A time dependent differential equation ẋ = f(t, x) ∈ R^{2} can be written as a timeindependent differential equation (ṫ, ẋ) = (1, f(t, x)) ∈ R^{3}. If t ↦ f(t, x) = f(t + T, x) is periodic in time t for all x ∈ R^{2}, we can introduce the two dimensional surface

Σ = ⋃_{n∈Z}{n · T } × R^{2} ⊂ R^{3} .

The return map to Σ is just the time-one map x(t) ↦ x(t + T).

9.8 The ”strange” attractor

An attracting set K for a map T : X ↦ X is a compact set K which is invariant under the map and which has the property that there exists an open neighborhood V of K such that ⋂_{n>0} T^{n}(V) = K. A flow-invariant subset K for a differential equation ẋ = f(x) is called an attracting set if it is an attracting set for the time-one map T : x(t) ↦ x(t + 1).

Example: we have seen that there is an ellipsioid E, which is mapped into is interiour under the time one map T of the Lorenz flow. The set K = ⋃_{n>0} T^{n}(\overline{E}) is an attracting set for the Lorenz flow.

Let ẋ = f(x) be a differential equation in R^{n}. Denote with T_{t} the time t map of the flow so that x(t) = T_{t}x(0). If U be a region in R^{n} we denote with V (t) = ∫_{T_{t}(U)} dx the volume of T_{t}(U).

Proposition 9.8.1

\frac{d}{dt}V (t) = ∫_{U(t)} div(f(x)) dx

Proof. The formula for changing variables of multiple integrals gives

V (t) = ∫_{T_{t}(U)} dx = ∫_{U} detDT_{t}(x) dx

where DT_{t} is the Jacobean of T_{t}. Because T_{t}(x) = x + f(x)t + O(t^{2}) for t → 0, we get

DT_{t}(x) = 1 + ∇f · t + O(t^{2}) .

For any matrix A we have

det(1 + A · t) = 1 + tr(A) · t + O(t^{2}) ,

where tr(A) = ∑_{i} A_{ii} is the trace of A. We get therefore

det(DT_{t}(x)) = 1 + trDT_{t}(x) + O(t^{2}) = 1 + divf(x) · t + O(t^{2}) .

Therefore

V (t) = ∫_{T_{t}(U)} dx = ∫_{U} 1 + divf(x) · t + O(t^{2}) , dx .

Differentiation with respect to t gives the claim. □

We apply this now to the Lorenz flow:

Proposition 9.8.2 Given any open bounded set U ⊂ R^{3} and with V (t) volume of its image T_{t}(U). Then

V (t) = V (0) · e^{−(σ+b+1)t} .

Proof. The divergence of the Lorenz flow (ẋ, ẏ, ż) = f(x, y, z) is

∂_{x}ẋ + ∂_{y}ẏ + ∂_{z}ż = −(σ + b + 1) .

Therefore \frac{d}{dt}V (t) = −(σ + b + 1)V (t). □

Corollary 9.8.3 The Lorenz flow can not have repelling equilibrium points (”sources”) or invariant two-dimensional tori.

Proof. In both cases, there would exist an invariant open set U of R^{3} which is mapped onto itself so that μ(T (U)) ≥ μ(U). □

Proposition 9.8.4 There exists a closed connected attracting set K of zero Lebesgue measure μ for the Lorenz flow.

Proof. We have seen that there exists a closed bounded ellipsoid such that T (E) ⊂ int(E). By the above proposition, we have μ(T^{n}(E)) ≤ λ^{n}μ(E) → 0 for n → ∞ and λ = e^{−(σ+b+1)} < 1. The set K := ⋂_{n} T^{n}(\overline{E}) is invariant under T and has zero Lebesgue measure. Because E is closed and connected, also T^{n}(E) is closed and connected and so K is closed and connected. □

The concept of attractor or strange attractor is not defined in a unique way. An attractor does not necessarily have to attract a whole neighborhood. Some people call an attracting set ”attractor”. It is however better to use an other definition for attractors since there are many important examples of ”attractors”, which are not attracting sets.

A widely accepted definiton of attractor is the one of Milnor: Let T : R → R be a continuous map and let ω^{+}(x) = ⋂_{m>0} O^{+}(T^{m}x) be the omega limit set of x. A set K ⊂ R^{n} is called an attractor, if ρ(K) = {x ∈ R^{n} | ω^{+}(x) ⊂ K} has positive Lebesgue measure and such that there exists no closed subset K′ of K such that ρ(K′) = ρ(K) up to a set of measure zero. The basin of attraction ρ(K) of K does not necessarily have to be open. One is mainly interested in minimal attractors, that is attractors for which no propoer subset is an attractor.

Strangeness is is not properly defined neither. A possible definition is a strange attracting set of a topological dynamical system (X, T) is an attracting set K of T such that the system (K, T) is chaotic in the sense of Deveney and such that K is a fractal. A strange attracting set of a differential equation ẋ = f(x) is a flow invariant compact subset which is a strange attractor for the time-one map T.

It is not known whether the Lorenz attractor is a strange attracting set in the above sense. For a discussion about the definition of attractors, we refer to [?].

9.9 Reconstruction of attractors from datas

Assume we have given a complicated dynamical system ẋ = f(x). Assume we can do measurements by getting data f(x) ∈ R. We may only probe the system at discrete time intervals and get the values f_{k} = f(x(k · τ)) of one of the state variables f(x(t)). If the dynamical system ẋ = f(x) has a finite dimensional attractor, we can hope that for some N

z_{k} = ϕ(z_{k−1}, …, z_{k−N})

holds. In this case, a map T : R^{N} ↦ R^{N} determines the dynamics of the whole complicated system. This method is called the phase space reconstruction of an attractor.

Evenso this construction seems to be quite hypothetical, there are results of Mané and Takens which tell that if the dimension of the attractor K is ≤ d ∈ N, then a reconstruction in dimension N = 2d+1 should work. Choosing the dimension 2d+1 of the embedding space makes it very probable that the projection from K to R^{N} is injective. The number 2d + 1 is related to a theorem of Whitney which says that any smooth connect closed manifold M of dimension d can be smoothly embedded in R^{2d+1} and that any continuous map M → R^{2d+1} can be approximated arbitrarily closely by smooth embeddings. The map, which assigns to an attractor K of dimension ≤ d the image of the map

x ↦ (f(T x), f(T^{2}x), …, , f(T^{2d+1}x))

is a continuous map from K to R^{2d+1}.

9.10 Measurements

Many measurements have been performed in the Lorenz system. Especially for the values σ = 10, b = 8/3 and with the parameter r. For r = 28, the topological entropy has been measured to be log(2) which indicates, that for such a parameter, there is a shift embedded. The Hausdorff dimension of the attractor is measured between 2.062 and W2.063 [?].

9.11 Some limiting cases

The case σ = 0. The case σ = 0 is interesting because the flow is then a linear integrable flow: because x(t) = x is constant, we have

( ẏ ; ż ) = ( \begin{matrix}−1 & −x \\ x & −b \end{matrix} ) ( \begin{matrix}y \\ z \end{matrix} ) + ( \begin{matrix}r · x \\ 0 \end{matrix} ) .

This is integrable: write X(t) = (x(t), y(t), z(t)) and

A = ( \begin{matrix}0 & 0 & 0 \\ r & −1 & −x \\ 0 & x & −b \end{matrix} ) .

We have then X(t) = exp(At) · X(0). The eigenvalues of A are λ_{1} = 0 and

λ_{2,3} = −(1 + b)/2 ± \sqrt{(1 + b)^{2}/4 − (b + x^{2})} .

The case r = ∞. In order to pass to the limit r → ∞, we define new variables ξ, η, ζ a new time τ and a new parameter ϵ by

ϵ = r^{−1/2},
x = ϵ^{−1} · ξ, y = ϵ^{−2}σ^{−1}η
z = ϵ^{−2}(σ^{−1}ζ + 1)
t = ϵ · τ .

The Lorenz equations

ẋ = σ(y − x)
ẏ = rx − y − xz
ż = xy − bz

go then into the equation

ξ′ = η − ϵσ · ξ
η′ = −ξ · ζ − ϵ · η
ζ′ = ξ · η − ϵ · b · (ζ + σ)

where ξ′, η′, ζ′ are the time derivatives with respect to τ. (Such a coordinate transformation can also be perforemd with the computer. This is especially helpful, if one has to try many cases in order to find the right scaling.

   r=epsilon^(-2);
   x=epsilon^(-1) xi;                  dx=epsilon^(-1) dxi;
   y=epsilon^(-2) sigma^(-1) eta;      dy=epsilon^(-2) sigma^(-1) deta;
   z=epsilon^(-2)(sigma^(-1) zeta+1);  dz=epsilon^(-2) sigma^(-1) dzeta;
   t=epsilon tau;                      dt=epsilon dtau;
   Simplify[epsilon (dx - sigma (y-x) dt)]
   Simplify[sigma epsilon^2 (dy -  (r x - y -x z) dt)]
   Simplify[sigma epsilon^2 (dz - (x y - b z) dt)]

The limit ϵ → 0 makes sense and the differential equations are then

ξ′ = η
η′ = −ξ · ζ
ζ′ = ξ · η .

This differential equation is ξ′′′ = −ξ^{2}ξ′ which gives after integration ξ′′ = −ξ^{3}/3 + C. The system has the integral ξ^{′2}/2 + ξ^{4}/12 − Cξ and ξ(t) is oscillating periodically.

We see that every orbit is periodic. One can show that these periodic solutions (ξ(τ), η(τ), ζ(τ)) are stable in some regions and exist also for small ϵ. This means that for large r, the Lorenz system has attracting periodic solutions in some parameter domains. For more information see Sparrow’s book [?].

9.12 Other systems

The Rössler system A differential equation

\begin{aligned}ẋ & = & −y − z \\ ẏ & = & x + ay \\ ż & = & b + xz − cz \end{aligned}

is called the Rössler system. For a = 0.2, b = 0.4, c = 5.7 the dynamical system models a chemical reaction. A famous example of such a chemical reaction is the Belousov-Shabotinsky system a system which may evolve chaotically. Interesting parameter values for the Rössler system are a = b = 0.2, and 2.5 < c < 10.0.

For c = 8.0, there exists a stable periodic solution.

Orb=NDSolve[{x'[t]==-(y[t]+z[t]),y'[t]==x[t]+.2y[t],z'[t]==.2+x[t]*z[t]-8.0z[t],
  x[0]==z[0]==0,y[0]==.3},{x,y,z},{t,500,700},MaxSteps->9000];
ParametricPlot3D[Evaluate[{x[t],y[t],z[t]} /. Orb],{t,500,700},
  PlotPoints->5000];

The ABC flow. The ABC flow is defined by

ẋ = A sin(z) + C cos(y)
ẏ = B sin(x) + A cos(z)
ż = C sin(y) + B cos(x)

where A, B, C are nonzero constants. Because the differential equations are invariant under x ↦ x + 2π, y ↦ y + 2π or z ↦ z + 2π, the differential equation is defined on T^{3}.

Time dependent Hamiltonian systems. Given a function V (t, x) which is periodic both in t and x. The differential equation ẍ = V (t, x) is equivalent to

\begin{aligned}ẋ & = & y \\ ẏ & = & V (t, x) \\ ṫ & = & 1 \end{aligned}

Also this differential equation is defined on T^{3}. (see exercices).

The Japanese attractor. The differential equation

ẍ + k · ẋ + x^{3} − γ · cos(t)

can be written as

\begin{aligned}ẋ & = & y \\ ẏ & = & −k · y − x^{3} + γ · cos(t) \\ ṫ & = & 1 . \end{aligned}

Taking the Poincaré map (x(0), y(0)) ↦ x(1), y(1) gives a map on R^{2}. For k = 0.1, γ = 12, one obtains the Ueda or Japaneese attractor.

9.13 Epilog

Why is the Lorenz system interesting?

• The Lorenz attractor is of historical interest. Lorenz used the model to justify the ’butterfly effect’ a metaphor of the imprecision of weather forecasting.

• It models a Galerkin approximation of Navier Stokes equations describing a horizontal fluid layer heated from below. It is an example for the hope that infinite dimensional systems can be modeled by finite dimensional ordinary differential equations.

• It is from physical relevance on the connection with the Maxwell-Bloch equations for lasers and on the convection problem in toroidal regions.

• It shows numerically a strange attractor.

• It leads on to a discussion of knots in dynamical systems because periodic orbits of flows in R^{3} form often knots.

• It is a simple three dimensional nonlinear system of autonomous ordinary differential equations. At the present, it is still almost the only candidate for a system having a well understood strange attractor.

• Almost all types of chaotic behavior can be found, for some parameter values in the Lorenz equations.

A little bit of history (see [56] ) Early in the twentieth century Lord Rayleight systematically studied convective currents in a region with constant depth. Barry Salzman derived in 1962 a system of differential equations that described convection. From that system Lorenz invents 1963 the differential equations bearing his name as a model of a two-dimensional convection. The paper of Lorenz was largely ignored for nearly ten years. One reason was that the Journal of Athmospheric sciences, in which he published his paper is not read so frequently by mathematicians and physicists. In the seventies, the interest in dynamical systems exploded and the notion of chaotic behavior appeared. Guckenheimer-Williams proposed 1980 a geometric Lorenz system which seems to have the same properties as the Lorenz equations though the connection depends on several global properties which have not yet been proved. Young 1981 showes that this geometric Lorenz system has a horse shoe and so positive topological entropy. For the real Lorenz system, there are now computerassisted proofs of homoclinic points. The book by C.Sparrow about the Lorenz equations contains the state of the art 1982. Newer articles with more references are amoung the references below.

9.14 Exercices

Exercice 9

Topic: Poincaré sections of differential equations and twist maps A differential equation ẋ = f(x) in R^{n} has a shift embedded, if the time one map x(t) ↦ x(t + 1) has a shift embedded. You will construct here differential equations in R^{3} which which have a shift embedded.

1) The differential equation ẍ = γ · f(x) · g(t) with smooth functions f and g is equivalent to the differential equation

ẋ = y
ẏ = γ · f(x) · g(z)
ż = 1

in R^{3}.

2) Show that

Σ = ⋃_{n} Σ_{n} = ⋃_{n} R^{2} × {2πn}

is a surface, for which the Poincaré map is defined for all points in Σ. Show that when g is 2π periodic, the Poincaré map T̃_{ϵ,δ,γ} : Σ → Σ which maps Σ_{n} to Σ_{n+1} can be described by one map T : R^{2} ↦ R^{2}.

3) Assume g(z) = g_{ϵ,δ}(z) is a family of real analytic periodic function which converge for ϵ → 0 to the Dirac distribution ∑_{n} δ(x + δ + 2πn). We mean with this that for any smooth function h

∫_{s_{1}}^{s_{2}} ∑_{n} h(x)g_{ϵ,δ}(x) dx → ∑_{s_{1}≤n≤s_{2}} h(−δ + 2πn) .

Show that for ϵ, δ → 0, the map T_{ϵ,δ,γ} converges to the twist map

T_{γ} : (x, y) ↦ (x + y + γf(x), y + γf(x)) .

4) Assume, {σ | f(σ) = 0, f′(σ) ≠ 0} contains N ≥ 2 points. Show that for γ large enough and ϵ, δ small enough, the map T_{ϵ,δ,γ} has a shift with N symbols embedded.

5) Find an open set of differential equations on the three dimensional torus T^{3} which have an embedded shift.

N-body problems

10.1 The celestial N body problem

The general N body problem. Consider N point masses q_{i} with mass m_{i} moving in R^{3}. Newton’s second law and the law of gravity yield to the equations

m_{j} · q̈_{j} = ∑_{j≠i} G · m_{i}m_{j} \frac{q_{i} − q_{j}}{|q_{i} − q_{j}|^{3}} = \frac{∂V}{∂q_{i}} ,

where V = ∑_{i<j} Gm_{i}m_{j} \frac{1}{|q_{i}−q_{j}|} is the potential. The constant G = 6.6732·10^{−11}N ·m^{2}·kg^{−2} is the gravitational constant. We can take units such that G = 1.

With M = Diag(m_{1}, … m_{N}), q = (q_{1}, …, q_{N}), the equations can be written shortly as

Mq̈ = \frac{∂V}{∂q} .

If we define p = Mq̇, we get the Hamiltonian system

q̇_{i} = \frac{∂H}{∂p_{i}} = \frac{p_{i}}{m_{i}} , ṗ_{i} = −\frac{∂H}{∂p_{i}} = −\frac{∂V}{∂q_{i}}

with the Hamiltonian

H(q, p) = ∑_{i} \frac{p_{i}^{2}}{2m_{i}} − V (q) = \frac{||p||^{2}}{2} − V (q) .

The phase space of the N body problem is the set

D = R^{3N} \ Δ ,

where Δ = ⋃_{1≤i≤j≤N}{(q, p) | q_{i} = q_{j}} is the set of collisions.

Proposition 10.1.1 The N-body problem has 10 classical integrals of motion:

a) The total momentum P = ∑_{i=1}^{N} p_{i}.

b) The velocity Ċ of the center of mass C = ∑_{i} q_{i}.

c) The total angular momentum L = ∑_{i} q_{i} × p_{i}.

d) The energy H.

Proof. a) Every term in the sum P appears twice with opposit sign.

b) MĊ = P. c)

L̇ = ∑_{i} q̇_{i} × p_{i} + ∑_{i} q_{i} × ṗ_{i} .

The second sum is zero because q_{i} × (q_{i} − q_{j}) = −q_{i} × q_{j} and because each term in the remaining sum appears twice with oppsit time. □

One would need 6N − 1 integrals of motion to integrate the N-body problem. The 10 classical integrals are far not enough if N > 2.

The existence theorem for differential equations implies that for any initial condition, for which all the particles are at different places, there exists a unique solution at least for some short time.

Proposition 10.1.2 (Painlevé) Let (q(t), p(t)) be a solution of the N body problem. Let (t_{1}, t_{2}) be the maximal interval, where the solution exists. If |t_{i}| < ∞ then |V (q(t))| → ∞ for t → t_{i}.

Proposition 10.1.3 (Zeipel) If some of the particles do not tend to a finite limiting position as t approaches t_{i}, then one has necessarily lim_{t→t_{i}} R(t) = ∞, where R = max_{i,j} r_{ij} is the maximum of the mutual distances.

If t_{i} is finite, the solution (q(t), p(t)) is said to approach a singularity. If there exists (q, p) ∈ Δ such that (q(t), p(t)) → (q, p) for t → t_{i}, then (q, p) is called a collision. The question about the existence of other singularities is called Painlevé’s problem. Zeipel’s theorem sais that in that case some particles have to go to infinity in finite time.

It was only established recently with 5 particles that other singularities exist. For the three body problem an other result of Painlevés assures that such a singularity can not happen. For the three body problem, all singularities are collisions.

It is still not known whether the set of initial configurations (in the 6N dimensional phase space) having global solutions, has full Lebesgue measure.

The restricted three body problem The restricted three body problem is a limiting case of the three body problem in which the mass of the third body is zero and in which the three bodies are in a plane. One assumes that the two massive bodies are moving in circular orbits around their center of mass.

It models a system with two large bodies and a small body. Examples of such systems in our solar system are: an asteroid moving in the Sun-Saturn system. Or the moon moving in the Sun-Earth potential.

The two first bodies move around each other periodically. The motion of the third infinitesimal particle is governed by the differential equation. In rotating coordinates, the two first bodies are at rest. The first body with mass μ has the coordinates (1 − μ, 0), the second body with mass 1 − μ has the coordinates (−μ, 0). The equations describing the system are

q̈ = 2Jq̇ + ∇V (q)

with

J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , V (q) = \frac{||q||^{2}}{2} + \frac{1 − μ}{r_{1}(q)} + \frac{μ}{r_{2}(q)} ,

where r_{1}^{2} = (q_{1} − 1 + μ)^{2} + q_{2}^{2}, r_{2}^{2} = (r_{1} + μ)^{2} + q_{2}^{2} are the distances from q to the two fixed bodies. The system has the integral

h = \frac{||q̇||^{2}}{2} − V (q)

called Jacobi integral which comes from the conservation of the energy of the motion of the third body. For fixed energy h, the motion of the third body takes place in the Hill region

{q ∈ R^{2} | V (q) + h ≥ 0} .

There are energy values, for which this region is bounded. (See Fig. 2).

The differential equations of the restricted three body problem can be written as a Hamilton system by introducing the momenta p_{1} = ẋ_{1} − x_{2} and p_{2} = ẋ_{2} − x_{1} and the Hamiltonian

H = \frac{||p||^{2}}{2} + (q, Jp) − V (q)

with

V (q) = \frac{μ}{r_{1}(q)} + \frac{1 − μ}{r_{2}(q)} .

Denote with T_{t} time-t map of the flow given by the Hamiltonian flow (q_{t}, p_{t}) ∈ R^{4}. The following theorem holds for general Hamiltonian systems.

Theorem 10.1.4 (Liouville)

For all t, the map T_{t} is area preserving.

Proof. Since the Jacobian dT_{t} is symplectic, it has determinant 1 and the map is area preserving. □

It shows that if we restrict the flow to an interval [a, b] of energies for which the Hill region is bounded, one can apply ergodic theory. Restriction to a three dimensional compact energy surface and taking a Poincaré map onto a two dimensional hyperplane gives a two dimensional area preserving map.

The Sitnikov problem. The Sitnikov problem is also a restricted three body problem. The configuration is the following. Two mass points of equal mass ove under Newton’s law of attraction in elliptic orbits while the center of mass is at rest. A third mass point of zero mass is moving on the line perpendicular to the plane of motion of the first two masses. Again, since the third mass point is not affecting the motion of the two primary particles.

If time is normalized such that the period of the primaries is 2π, if the masses are normalized such that m_{1} = m_{2} = 1/2 and the length is normalized such that the gravitational constant satisfies G = 1, the differential equation is

z̈ = −\frac{z}{(z^{2} + r^{2}(t))^{3/2}}

with r(t) = \frac{1}{2}(1 − ϵ · cos(t)) + O(t^{2}). The parameter ϵ is called the excentricity.

Consider a solution z(t) which has infinitely many zeros t_{k} fo z which are ordered according to size t_{k} < t_{k+1}. The integers

s_{k} = [\frac{t_{k+1} − t_{k}}{2π}]

measure the number of complete revolutions of the primaries between two zeros of z(t). To every such solution one can assign a doubly infinite sequence of integers.

Theorem 10.1.5 (Sitnikov,Moser) For ϵ > 0 small enough, there exists m ∈ Z, such that any sequence s ∈ Z^{Z} with s_{k} ≥ m corresponds to a solution of the above differential equation.

By taking an unbounded sequence s_{k}, one gets solutions which are unbounded and which return again and again to zero. The proof of this theorem can be found in [48].

The planar isosceles three body problem The planar isosceles problem is a special case of the planar three body problem which reduces to a Hamiltonian system of two degrees of freedom. One takes two equal masses m_{1}, m_{2} whose initial position and velocity are symmetric with respect to the y-axis in the plane and a third mass whose initial position and velocity lie along this axis. Because of the symmetry of the problem, the masses will remain in such a configuration for all time. After fixing the center of mass at the origin, one obtains a system with two degrees of freedom. Let x_{1} denote the distance between m_{1} and m_{2} and let q_{2} denote the distance between m_{3} and the center of mass of m_{1} and m_{2}, The differential equations for x_{1}, x_{2} are

q̈_{1} = \frac{−2m_{1}}{q_{1}^{2}} − \frac{8m_{3}q_{1}}{(q_{1}^{2} + 4q_{2}^{2})^{3/2}}
q̈_{2} = \frac{−8(2m_{1} + m_{3})q_{2}}{(q_{1}^{2} + 4q_{2}^{2})^{3/2}}

With the matrix M = Diag(m_{1}/2, 2m_{1}m_{3}/(2m_{1} + m_{3})), one can introduce the momentum p = Mq and write the system as a Hamiltonian system with Hamiltonian

H(q, p) = \frac{1}{2}(p, M^{−1}p) + V (q)

with

V (q) = −\frac{m_{1}^{2}}{q_{1}} − \frac{4m_{1}m_{3}}{(q_{1}^{2} + 4q_{2}^{2})^{1/2}} .

See [14] for the problem of triple collisions for this system.

10.2 Vortex system

The motion of N vortices z_{i} with vorticity ω_{i} in the complex plane is given by the differential equations

ω_{k} · \frac{d}{dt}\overline{z_{k}} = \frac{1}{2πi} ∑_{j≠k} \frac{ω_{j}}{z_{k} − z_{j}}

which is a Hamiltonian system

ω_{k}ẋ_{k} = \frac{∂H}{∂y_{k}} , ω_{k}ẏ_{k} = −\frac{∂H}{∂x_{k}}

for the variables x_{i} + iy_{i} = z_{i} with the Hamiltonian

H(x, y) = −\frac{1}{4π} ∑_{i≠j} ω_{i}ω_{j} log(|z_{i} − z_{j}|) = −\frac{1}{2π} ∑_{i<j} ω_{i}ω_{j} log(|z_{i} − z_{j}|) .

One can also write shorter

ω_{k} · \frac{d}{dt}\overline{z}_{k} = \frac{2}{i} \frac{∂H}{∂z_{k}}

with

\frac{∂}{∂z_{k}} = \frac{1}{2}(\frac{∂}{∂x_{k}} − \frac{∂}{∂y_{k}}) .

The phase space of the system is

D = C^{N} \ Δ ,

where Δ = ⋃_{1≤i≤j≤N}{z_{i} = z_{j}} is the set of collisions.

Numerical experiments With the following program, we can integrate the motion of arbitrary many vortices in the plane. It is assumed that the vorticity is constant ω_{i} = 1.

Ste[f_,y_,y0_, dt_]:=Module[{k1,k2,k3,k4},
  k1=dt N[f /. Thread[y->y0]     ];k2=dt N[f /. Thread[y->y0+k1/2]];
  k3=dt N[f /. Thread[y->y0+k2/2]];k4=dt N[f /. Thread[y->y0+k3  ]];
  y0+(k1+2*k2+2*k3+k4)/6];
Las[f_,y_,y0_,t_,dt_]:=Module[{q=y0},Do[q=Ste[f,y,q,dt],{i,Round[t/dt]}];q];
Var[n_]:=Table[z[i],{i,n}]; e=0.0001; m[i_,n_]:=Mod[i-1,n]+1; a=N[1/(2 Pi I)];
Dif[n_]:=Table[a*Sum[w=z[i]-z[m[j,n]];w/(Abs[w]^2+e),{j,i+1,i+n-1}],{i,n}];
Vor[c0_,t_,dt_]:=Module[{l=Length[c0],c},c=Var[l];h=Dif[l];Las[h,c,c0,t,dt]];
Ini[n_]:=Module[{s=2*Pi/n},N[Table[Exp[I*s*j],{j,n}]] ];
Pic[z_]:=ListPlot[Table[{Re[z[[k]]],Im[z[[k]]]},{k,Length[z]}],
  PlotRange->{{-3,3},{-3,3}},Axes->False];
Mov[P_,K_,T_]:=Module[{r=Ini[P]},Do[Pic[r];r=Vor[r,T,0.01],{K}]];

The motion of one of the vortices can be seen with the following procedures:

Orb[f_,y_,y0_,t_,d_]:=Module[{q=y0,s={}},
  Do[q=Ste[f,y,q,d];s=Append[s,q],{Round[t/d]}];s]
Vortex[n_]:=Orb[Dif[n],Var[n],Ini[n],2.0,0.01];
Pro[n_]:=Module[{z=Vortex[n]},Table[{Re[z[[i,1]]],Im[z[[i,1]]]},{i,Length[z]}]];
Show[Graphics[{Line[Pro[4]]}],Frame->True,Axes->False]

Proposition 10.2.1

The N vortex system has 4 classical integrals of motion:

a) the center of mass Z = Q + iP = ∑_{i} ω_{i}z_{i}

b) the angular momentum I = ∑_{i} ω_{i}|z_{i}|^{2}.

c) the energy H.

Proof. a) Every term in ∑_{i} ω_{i}ż_{i} appears twice with oppsite sign.

b) Define

S = ∑_{k} ω_{k}z_{k}\overline{ż}_{k}

Because I and İ are real, we have İ = S + \overline{S}.

Ṡ = \frac{1}{2πi} ∑_{k≠j} \frac{ω_{k}ω_{j}z_{k}}{z_{k} − z_{j}} = \frac{1}{2πi} ∑_{k≠j} ω_{k}ω_{j}

is purely imaginary and so İ = S + \overline{S} = 0.

c) Ḣ = ∑_{i} H_{x_{i}} ẋ_{i} + H_{y_{i}} ẏ_{i} = ∑_{i} H_{x_{i}}H_{y_{i}}ω_{i}^{−1} − H_{y_{i}}H_{x_{i}}ω_{i}^{−1} = 0. □

These integrals come from the invariance of the differential equations by space translations, space rotations and time translations.

Define the Poisson bracket

\{f, g\} = ∑_{i=1}^{N} \frac{1}{ω_{i}} (\frac{∂f}{∂x_{i}} \frac{∂g}{∂y_{i}} − \frac{∂f}{∂y_{i}} \frac{∂g}{∂x_{i}})

for two smooth functions f, g : C^{N} → C. Using the notation ∂_{z} = \frac{∂}{∂z} = \frac{1}{2}(∂_{x} − i∂_{y}) and the scalar product < u, v >= Re(u · \overline{v}) on C, we can also write

\{f, g\} = ∑_{i} \frac{4}{ω_{i}} < ∂_{z_{i}}f, i∂_{z_{i}}g > .

The equations of motion are then ż_{i} = {z_{i}, H} or

ẋ_{i} = {x_{i}, H}, ẏ_{i} = {y_{i}, H} .

Because the integral are invariant under the motion, one has

{I, H} = {Z, H} = 0 .

Of course one can build also other integrals by combining the known ones. So, also

J := \frac{1}{2}|Z|^{2} = \frac{1}{2}(P^{2} + Q^{2})

is an integral.

One can show that H, I, J have pairwise vanishing Poisson bracket. Indeed, this implies that a system of three vortices is integrable. The explicit integration can be found for example in [?]. The threshold for chaotic behavior in a point-vortex system starts with N = 4. Indeed already the restricted four vortex system, where the vortex strenght of the forth vortex is zero is already is not integrable [62]. The fourth vortex is in this case just a particle of the fluid.

Proposition 10.2.2 If all the vorticities ω_{i} > 0, and z ∈ D, then the solution of the vortex flow exists for all times.

Proof. In the case when all ω_{i} > 0, all the solutions in the phase space are bounded, because

I = ∑_{i} ω_{i}|z_{i}|^{2} = const

defines a compact set. The flow is then defined for all times since H = const prevents collisions. □

In general, when ω_{i} can take both signs, the existence and uniqueness problem is nontrivial since the logarithmic divergence of the Green function may generate catastrophes.

There are initial conditions of a three vortex situation leading to a collapse in finite time.

Let z_{1} = (−1, 0), z_{2} = (1, 0) be the initial position of two vortices with vorticity ω_{1} = ω_{2} = 2 and z_{3} = (1, \sqrt{2}) be the initial position of a vortex with vorticity ω_{3} = −1. Write a_{i} for the length of the sides of the triangle formed by three vortices and denote with A its area. One has

\frac{d}{dt}a_{k}^{2} = \frac{2ω_{k}A}{π}(\frac{1}{a_{k−1}^{2}} − \frac{1}{a_{k+1}^{2}} ) .

The ratios of the sides are conserved and \frac{d}{dt}a_{k}^{2} = −\frac{1}{3\sqrt{2}π} a_{k}^{2}(0). This gives explicitely

a_{k}(t) = a_{k}(0)\sqrt{1 − \frac{t}{3\sqrt{2}π}}

and a collaps at time t = 3\sqrt{2}π.

Remarks. Using KAM theory Khanin showed that in the phase speac of any system with an arbitrary number of vortices, there exists a set of initial conditions of positive measure for which the motion of vortices is quasi-periodic. Celletti and Falcolini have even quantitive results for the break down of invariant tori in the four vortex problem.

The restricted four vortex problem The three vortex problem is integrable and one expects chaos for the motion of four vortices. The restricted four vortex problem is obtained by taking an integrable three vortex motion and letting evolve a fourth vortex of zero vorticity in the stream lines of the others.

The next lemma part c) is due to Novikov-Sedov 1978,Aref-Pomphrey:

Lemma 10.2.3 a) {I, H} = {P, H} = {Q, H} = 0

b) {I, Q} = −P, {I, P } = Q}

c) {I, J} = 0

Proof. a)

\frac{d}{dt}(P + iQ) = \frac{1}{2πi} ∑_{j≠k} \frac{ω_{j}}{z_{k} − z_{j}}

changes sign when replacing j with k. Because the sum does not change by this permutation, it must vanish. To the Hamiltonian I belongs the differential equation

\frac{d}{dt}\overline{z}_{k} = \frac{∂}{∂z_{k}} I = \overline{z}_{k}

which induces a rotation z_{k}(t) = e^{it}z_{k}. Because the Hamiltonian H is invariant under this flow, we have {H, I} = 0. This gives also {I, H} = 0 which means that I is an integral.

b) We get ∂_{z_{i}}Z = ω_{i}, ∂_{z_{i}}I = \frac{ω_{i}}{2} \overline{z}_{i} and so

∑_{i} \frac{4}{ω_{i}} < ω_{i}, iω_{i}z_{i} >= −iZ

giving {I, P } = Q, {I, Q} = −P. □

The motion of vortices in arbitrary domains. Let D be a 2-dimensional manifold which can have a boundary. Let g(x, y) be the Green function of the Laplacian with Dirichlet boundary conditions. This means that g(x, y) solves the Poisson equation Δ_{x}g(x, y) = δ(y) and g(x, y) = 0 for x ∈ δD or y ∈ δD and where Δ_{x} is the Laplacian on the manifold.

Write z_{i} = x_{i} + iy_{i} for the local coordinates of a point vortex on D and write J∇_{i} = 2∂_{z_{i}} = ∂_{x_{i}} − i∂_{y_{i}}.

The motion of the vortices is defined by the differential equation

ω_{i} \frac{d}{dt}\overline{z}_{i} = J∇_{i}H = J∇_{i} ∑_{i≠j} ω_{i}ω_{j}g(z_{i}, z_{j}) .

With ω = ω_{1}, ω_{2}, … ω_{N}), z = (z_{1}, …, z_{N}) and ∇ = (∇_{1}, …, ∇_{N}) and H(z) = ∑_{i≠j} ω_{i}ω_{j}g(z_{i}, z_{j}), this can be written shortly as ωż = J∇H(z).

Examples:

(1) On the plane D = R^{2}

g(x, y) = −\frac{1}{2π} log |x − y|

(2) On the torus D = T^{2} = R^{2}/(2L)^{2} one has

g(x, y) = −\frac{1}{(2πL)^{2}} ∑_{k∈Z^{2}\\{0\}} e^{\frac{2πi}{L} k·(x−y)} .

One must require ∑_{i} ω_{i} = 0.

(3) On the cylinder D = R × T with T = R/(2L) one has

g(x, y) = −\frac{1}{2π} log sin(\frac{π}{L}(x − y)) .

Also here, one requires ∑_{i} ω_{i} = 0. Famous is the Karman vortex street, where one has 2 vortices of opposite vorticity.

(4) On the sphere D = S^{2} one gets

g(x, y) = log(1 − x · y) ,

where one writes x, y ∈ S^{2} ⊂ R^{3} with the scalar product x · y.

(5) On the two dimensional disc D = D = {|z| ≤ R} one gets (see [?])

g(x, y) = −\frac{1}{2π} log(\frac{|x − y|R}{|y||x − \overline{y}|})

where \overline{y} = R^{2}y/|y|^{2} is the conjugate point of y. For the Hamiltonian one has then

H = −\frac{1}{4π} (∑_{i≠j} log(\frac{|z_{i} − z_{j}|R}{|z_{i}||z_{i} − \frac{z_{j}}{|z_{j}|^{2}} |}) + ∑_{i} ∑ log(1 − |z_{i}|^{2}) ,

where the second sum is the energy of the selfinteraction of the vortices with its mirror vortices.

For general vortex strengths, vortices can collide. Dürr and Pulvirenti proved in 1982 that the set of initial conditions leading to the collaps of two or more vortices has Lebesgue measure zero provided the vortices move in a bounded domain or on the torus.

10.3 The Toda system

The N particles systems we considered until now had the property that each particle was interacting with each other particle. The differential equation was of the form

M · q̈ = ∇V (q)

with

V (q) = ∑_{i≠j} f(q_{i} − q_{j}) .

If the N particles have only a nearest neighbor interaction and are located on a chain, the potential has the form

V (q) = ∑_{i} f(q_{i} − q_{i−1}) .

Consider a chain of particles q_{n} = q_{n+N} with mass m_{i} = 1 and with potential f(q) = e^{−q}. The motion of these particles is given by the differential equations

\frac{d^{2}}{dt^{2}} q_{n} = e^{q_{n+1}−q_{n}} − e^{q_{n}−q_{n−1}} ,

where the indices are taken modulo N. This system is called the periodic Toda system.

After a coordinate transformation

4a_{n}^{2} = e^{q_{n+1}−q}, 2b_{n} = p_{n} ,

these equations

\begin{aligned}q̇_{n} & = & p_{n} , \\ ṗ_{n} & = & e^{q_{n+1}−q_{n}} − e^{q_{n}−q_{n−1}} \end{aligned}

go into

ȧ_{n} = a_{n}(b_{n+1} − b_{n})
ḃ_{n} = 2(a_{n}^{2} − a_{n−1}^{2}) .

Given a_{n}, b_{n}, we define the matrices

L = ( \begin{matrix}b_{1} & a_{1} & 0 & · & 0 & a_{N} \\ a_{1} & b_{2} & a_{2} & · & · & 0 \\ 0 & a_{2} & · & · & · & · \\ · & · & · & · & a_{N−2} & 0 \\ 0 & · & · & a_{N−2} & b_{N−1} & a_{N−1} \\ a_{N} & 0 & · & 0 & a_{N−1} & b_{N} \end{matrix} )

B = ( \begin{matrix}0 & a_{1} & 0 & · & 0 & −a_{N} \\ −a_{1} & 0 & a_{2} & · & · & 0 \\ 0 & −a_{2} & · & · & · & · \\ · & · & · & · & a_{N−2} & 0 \\ 0 & · & · & −a_{N−2} & 0 & a_{N−1} \\ a_{N} & 0 & · & 0 & −a_{N−1} & 0 \end{matrix} ) .

A matrix L is called a Jacobi matrix. We met them already in twist maps like billiards.

Lemma 10.3.1

The Toda differential equations can be written as L̇ = [B, L] = BL − LB .

Proof. It is convenient to define the matrices

(Au)_{n} = a_{n}u_{n+1}, (A^{∗}u)_{n} = a_{n−1}u_{n−1}, (Cu)_{n} = b_{n}u_{n} ,

where the indices are taken modulo N. Then L = A + A^{∗} + C and B = A − A^{∗}. We get BL−LB = (A−A^{∗})(A+A^{∗}+C)−(A+A^{∗}+C)(A−A^{∗}) = (AC−CA)+(CA^{∗}−A^{∗}C)+2AA^{∗}−2A^{∗}A which gives

Ȧ = (AC − CA), Ċ = 2AA^{∗} − 2A^{∗}A .
□

The equation

L̇ = [B, L] = BL − LB

is called a Lax pair. A fundamental property of such an equation is that it describes an isospectral deformation of the matrix L. A matrix is called unitary, if Q^{∗}Q = 1 , where Q^{∗} is the adjoint of Q.

Lemma 10.3.2

There exists a unitary matrix Q(t) such that

L(t) = Q(t)^{∗}L(0)Q(t) .

Proof. Consider the differential equation Q̇ = −BQ with Q(0) = 1.

(i) Q(t) is unitary: i.e. it satisfies Q(t)Q(t)^{∗} = 1. Proof. This is true for t = 0. Using B^{∗} = −B we get

\frac{d}{dt}Q(t)Q(t)^{∗} = −BQQ^{∗} − QQ^{∗}B^{∗} = −B − B^{∗} = B − B = 0

this is also true for all t > 0. (ii) Q(t)^{∗}L(t)Q(t) = L(0). Proof. Differentiation of this equation with respect to time on both hand sides gives

Q̇^{∗}LQ + Q^{∗}L̇Q + QLQ̇ = −Q^{∗}B^{∗}LQ + Q^{∗}(BL − LB)Q − Q^{∗}LBQ = 0 .
□

Corollary 10.3.3 The eigenvalues of L are invariant under the flow. Also the functions F_{n} = \frac{1}{n}tr(L^{n}) are integrals of the Toda flow.

Proof. The matrix L(t) is unitarily equivalent to L(0). It has therefore the same spectrum and all the eigenvalues of L are integrals of the Toda system. Because

tr(L(0)) = tr(Q^{∗}(t)L^{n}(t)Q(t)) = tr(L^{n}) ,

these functions are invariant. □

There is an other integral ∑_{n} log(a_{n}) of the flow because differentiation of ∑_{n} log(a_{n}) gives ∑_{n}(b_{n+1} − b_{n}) = 0. This integral can not be written as a trace.

Corollary 10.3.4

The solutions of the Toda system exist for all times.

Proof. tr(L^{2}) = ∑_{n} 2a_{n}^{2} + b_{n}^{2} stays finite. This is only possible, if a_{n}, b_{n} stay finite. □

The explicit integration of the periodic Toda lattice needs tools from algebraic geometry if a_{i} > 0 for all i and given in the next section.

There are other versions of Toda systems. In the case, when q_{0}(t) = −∞ and q_{N}(t) = ∞, the system is given by the differential equation

\frac{d^{2}}{dt^{2}} q_{n} = e^{q_{n+1}−q_{n}} − e^{q_{n}−q_{n−1}} .

In this case a_{0} = a_{N} = 0 and the differential equations can be written as L̇ = [B, L] where

L = ( \begin{matrix}b_{1} & a_{1} & 0 & · & 0 & 0 \\ a_{1} & b_{2} & a_{2} & · & · & 0 \\ 0 & a_{2} & · & · & · & · \\ · & · & · & · & a_{N−2} & 0 \\ 0 & · & · & a_{N−2} & b_{N−1} & a_{N−1} \\ 0 & 0 & · & 0 & a_{N−1} & b_{N} \end{matrix} )

and

B = ( \begin{matrix}0 & a_{1} & 0 & · & 0 & 0 \\ −a_{1} & 0 & a_{2} & · & · & 0 \\ 0 & −a_{2} & · & · & · & · \\ · & · & · & · & a_{N−2} & 0 \\ 0 & · & · & −a_{N−2} & 0 & a_{N−1} \\ 0 & 0 & · & 0 & −a_{N−1} & 0 \end{matrix} ) .

The Jacobi matrix L has only simple eigenvalues. In this case, the integration of the differential equation is easier.

Infinite particle systems with Toda interaction can be treated in the same way. The differential equation

\frac{d^{2}}{dt^{2}} q_{n} = e^{q_{n+1}−q_{n}} − e^{q_{n}−q_{n−1}}

is then defined in a Banach space of sequences q_{n}. This infinite-dimensional Hamiltonian system can be written as a deformation of infinite dimensional Jacobi matrices

(Lu)_{n} = a_{n}u_{n+1} + q_{n−1}u_{n−1} + b_{n}u_{n} .

10.4 Integration of the periodic Toda lattice

We want to give here an integration of the periodic Toda lattice in the case when a_{i} > 0 for all i. This section is more difficult but we try to be as selfcontained as possible.

We have seen, that the periodic Toda lattice can be written as a differential equation for a periodic Jacobi matrix L. This matrix defines an infinite dimensional tridiagonal operator

(L′u)_{n} = a_{n}u_{n+1} + a_{n−1}u_{n−1} + b_{n}u_{n}

on l^{2}(Z).

We say, a closed interval I in R is a spectral gap of the operator L′ if the boundary of I belongs to the spectrum and the interior is disjoint from the spectrum. There are finitely many spectral gaps for L′ and we index them arbitrarily (I_{n})_{n∈N}.

The Titchmarsh-Weyl functions m^{±} are meromorphic functions outside the spectrum of L′ defined as follows. Write X = {1, …, N} and let T be the cyclic permutation on X. Given x ∈ X, we define [L(x)]_{ij} = [L′(x + n)]_{ij}. We write also a(x) = a_{x} and b(x) = b_{x}.

Fix an energy E outside the spectrum of L′. The time-independent Schrödinger equation L(x)u = Eu admits a two dimensional family of solutions {u_{n}(x)} ∈ R^{Z}. If we fix for example u_{0}, u_{1}, all the other values u_{n} can be calculated recursively by

a_{n}u_{n+1} + a_{n−1}u_{n−1} + b_{n}u_{n} = Eu_{n} ,

where a_{n} = a(T^{n}x) and b_{n} = b(T^{n}x). With the vector

w_{n}(x) = (a_{n}(x)u_{n+1}(x), u_{n}(x)) ,

the Schrödinger equation can be written as the first order system

A_{E}(x)w_{−1}(x) = w_{0}(x) ,

where A_{E} is the transfer cocycle

A_{E}(x) = a^{−1}(T^{−1}x) ( \begin{matrix}E − b(x) & −a^{2}(T^{−1}x) \\ 1 & 0 \end{matrix} ) .

For E outside the spectrum of L, the cocycle A_{E} has a positive Lyapunov exponent: the Thouless formula

Re(tr(log(L − E))) = ∑_{i=1}^{N} log(a_{i}) + λ(A_{E})

shows that the Lyapunov exponent E → λ(A_{E}) is harmonic outside the spectrum. Because det(A_{E}(x)) = 1, the Lyapunov exponent takes values ≥ 0. According to the maximum principle for harmonic functions, the minimum 0 can not occur in the resolvent set. It follows from the multiplicative ergodic theorem (see [?]) that for E in the resolvent set, there exist one dimensional co-invariant stable and unstable vector spaces W^{±}(x) such that

A_{E}(x)W^{±}(x) = W^{±}(T x) .

For almost all x ∈ X we can take a unit vector w^{±}(x) ∈ W^{±}(x) and define u^{±}(x) as the second coordinate of w^{±}(x). Like this, there exist solutions u_{n}^{+}, u_{n}^{−} ∈ R^{Z} of L(x)u = Eu satisfying {u_{n}^{+}(x)} ∈ l^{2}(N) and {u_{n}^{−}(x)} ∈ l^{2}(−N). (The sequence u_{n} is determined by defining u_{0} as the second coordinate of w^{±}(x) and a(x)u_{1} as the first coordinate of w^{±}(x). The other entries u_{n} are then defined by L(x)u = Eu.) The Titchmarsh-Weyl functions are

m^{+}(x) = a(x)\frac{u^{+}(T x)}{u^{+}(x)} , m^{−}(x) = a(x)\frac{u^{−}(T x)}{u^{−}(x)} ,
n^{+}(x) = a(T^{−1}x)\frac{u^{+}(T^{−1}x)}{u^{+}(x)} , n^{−}(x) = a(T^{−1}x)\frac{u^{−}(T^{−1}x)}{u^{−}(x)} .

There can occur poles in each spectral gap. Define

M := m^{+} + m^{−}

and

G = (m^{+} − m^{−})^{−1} .

Because m^{−} − m^{+} is Herglotz, also −G^{−1} is Herglotz as the sum of two Herglotz functions and so G is a Herglotz function. We can write the functions M, G in the representation

M = ∫ \frac{dσ(E′)}{E − E′} , G = ∫ \frac{dg(E′)}{E − E′} ,

where dg is a measure and dσ is a signed measure. We see from the definitions that a pole of M corresponds to a zero of G.

Lemma 10.4.1 In the interior of a spectral gap I_{n}, the function M := m^{+} + m^{−} has one or no pole.

Proof. The Green function G is analytic in a spectral gap and the claim follows from the monotonicity of G which is

\frac{d}{dE} G > 0 .

An existing pole of m^{+} or m^{−} corresponds therefore to a zero of G and must be simple. □

We call μ_{n} the (when existing) pole of M = m^{+} + m^{−} in the gap I_{n}. Denote it with μ_{n}^{+} if it is a pole of m^{+} and μ_{n}^{−} if it is a pole of m^{−}.

Lemma 10.4.2 The Titchmarsh-Weyl functions m, n satisfy the following differential equations:

ṁ = 2m(n − n(T)) ,
ṅ = 2n(m(T^{−1}) − m) .

Proof. Verify that L̇ = [B, L] gives u̇ = B(t)u for any sequence u_{n}. The formulas follow now from the definition of the Titchmarch- Weyl functions. □

We can now determine the motion of μ_{n}^{±} under the first Toda flow.

Lemma 10.4.3

If μ_{n} is in the interior of I_{n}, then

μ̇_{n} = −2 · Res(M, μ_{n}) .

Furthermore μ̇_{n}^{+} > 0 and μ̇_{n}^{−} < 0.

Proof. From the formula mn(T) = a^{2} follows that a pole μ of m is a zero of n(T). We have therefore

0 = \frac{d}{dt}n(T) = \frac{d}{dE} n(T)μ̇ + ṅ(T)(μ)

and so

μ̇ = −\frac{ṅ(T)(μ)}{\frac{d}{dE} n(T)} = −2\frac{n(T)(m − m(T^{−1})}{\frac{d}{dE} n(T)}(μ)
= −2a^{2}(\frac{d}{dE} n(T))^{−1}(μ) .

Because m^{+} ≠ m^{−} inside a gap we have Res(M, μ) = Res(m, μ), where m is either m^{+} or m^{−}. The claim follows from

Res(M, μ) = Res(m, μ) = Res(mn(T)/n(T), μ) = Res(a^{2}/n(T), μ)
= a^{2}Res(\frac{1}{n(T)}, μ) = a^{2}(\frac{d}{dE} n(T))^{−1} ,

where we used the fact that μ is a simple zero of n(T). Because m^{−} and −m^{+} are Herglotz functions, we have

Res(m^{+}, μ) < 0, Res(m^{−}, μ) > 0 .
□

We now find explicit formulas for G^{−1} = m^{+} − m^{−}, M = m^{+} + m^{−}. It is known that L(x) has band spectrum

σ(L(x)) = ⋃_{i=1}^{g+1}[λ_{2i−1}, λ_{2i}] .

and that there exists g ≤ N − 1 gaps in the spectrum. Define

R(E) = ∏_{i=1}^{2g+2}(E − λ_{i}), l(E) = ∏_{i=1}^{g}(E − μ_{i}) ,

where μ_{i} denote as before the zeros of G.

Lemma 10.4.4

a) For n = 1, … g,

Res(M, μ_{n}^{±}) = Res(m^{±}, μ_{n}) = \frac{±\sqrt{R(μ_{n})}}{l′(μ_{n})} .

G^{−1} = m^{+} − m^{−} = \frac{\sqrt{R}}{l} .

b)

dσ = ∑_{i=1}^{g} Res(M, μ_{i})δ(μ_{i}) = ∑_{i=1}^{g} \frac{±\sqrt{R(μ_{n})}}{l′(μ_{n})}δ(μ_{i}) .

Proof. a) From the discrete Ricatti equation m(T) = E −b(T)−a^{2}/m and m(T^{N}x) = m(x) we get a continued fraction expansion

m = m(T^{N}) = E − b(T^{N}) − \frac{a^{2}(T^{N−1})}{E − b(T^{N−1}) − \frac{a^{2}(T^{N−2})}{⋯ \frac{⋯}{−b(T^{2}) − \frac{a^{2}(T)}{E − b(T) − \frac{a^{2}}{m}}}}} .

It follows that the functions m^{±} satisfy a quadratic equation. We can write the solutions as

m^{±} = \frac{A ± \sqrt{B}}{C} ,

where A, B, C are polynomials in E. Therefore

G^{−1} = m^{+} − m^{−} = 2\frac{\sqrt{B}}{C} ,
M = m^{+} + m^{−} = 2\frac{A}{C} .

Because m^{+} − m^{−} has zeros in λ_{i} and simple poles at the places μ_{i}, we get that R divides B and l divides C. We know

G^{−1} = m^{+} − m^{−} = O(E), E → ∞

and that there are maximal g zeros of l. One has at least 2g + 2 zeros of C because each of the λ_{i} is a zero of C. Maximally 2g + 2 zeros of C are possible because else the growth rate of G^{−1} at infinity would be too big. This implies B = c_{1} · R and C = c_{2} · l for two constants c_{1}, c_{2}. From the growth rate of G^{−1} = 2\sqrt{B}/C at infinity we get B = 4R, C = l. Since

m(E) = O(E), E → ∞ ,

we must have exactly g zeros of G which are poles of M if μ is in the interior of a gap.

b) From

G^{−1} = \frac{\sqrt{R(E)}}{l(E)}

we deduce

Res(M, μ) = \frac{\sqrt{R(μ)}}{l′(μ)} .

□

The next lemma shows that the integral of the measure dσ with respect to the time of the first Toda flow is absolutely continuous with respect to Lebesgue measure. This follows from the fact that the absolute velocity of a point in the support of dσ is twice the weight of the point.

Lemma 10.4.5

There exists h(t) ∈ L^{∞}(I, 2N) such that

dρ(t) := ∫_{0}^{t} dσ(s) ds = h dE .

Proof. Assume μ_{i}(0) is in the interior of a spectral gap. For small enough time t_{0}, the point μ_{i}(t_{0}) is still in the gap and has not yet reached the boundary of the gap. Because the velocity of the point μ is equal to the weight of the measure

−2 · Res(M, μ)δ(μ) ,

we get

∫_{0}^{t_{0}} −2 · Res(M, μ(s))δ(μ(s)) ds = −2 · 1_{[μ(0),μ(t_{0})]} dE .

If the point μ reaches the boundary of the gap, the weight of the point measure changes sign and moves to the other side. Because also the sign of R(μ) changes, the sign of dσ(t) does not change. The function h(E, t)/2 is equal to the number of times μ has passed the point E in the time-interval [0, t]. □

Define for n = 0, …, g − 1 the functions

f_{n} = \frac{E^{n}}{\sqrt{R}} .

They are in L^{1}(I) and form a basis in the space of differentials of the first kind on the hyperelliptic curve given by y = \sqrt{R}. All the functions f_{n} are in L^{1}(R, dE). We conjecture that this stays true in the general random case.

Lemma 10.4.6

The evolution of the first Toda flow can be written as

(f_{n}, h(t)) = ∫ f_{n}h(t) dE
= \{ \begin{matrix}(f_{n}, h(0)) + 2t & for n = g − 1 \\ (f_{n}, h(0)) & else. \end{matrix}

Proof.

\frac{d}{dt}(f_{n}, h) = ∫_{I} f_{n} dσ = ∑_{i=1}^{g} \frac{μ_{i}^{n}}{\sqrt{R(μ_{i})}}μ̇_{i}

= ∑_{i=1}^{g} \frac{2μ_{i}^{n}}{l′(μ_{i})} = \frac{2}{2πi} ∫_{Γ} \frac{E^{n}}{l(E)} dE

= \{ \begin{matrix}2 & for n = g − 1 \\ 0 & else. \end{matrix} ,

where Γ is a curve in the complex plane which goes once around the spectrum of L. □

Lemma 10.4.7 (Jacobi)

The mapping

{μ_{n}}_{n=1}^{g} ↦ (f_{n}, h(t))_{n=1}^{g}

is invertible.

Proof. In the usual language this means that the Abel-Jacobi map

\{μ_{i}\} ↦ ∑_{i=1}^{g} ∫_{γ_{2i}}^{μ_{i}} \frac{E^{n}}{\sqrt{R(E)}} dE

is invertible. For a proof see [?] Theorem 1, p.56. □

The density of states can be written as ∫_{X} dg(x) = dk. The knowledge of dk determines the points λ_{n}. From λ_{n} and μ_{n} the operator L can be reconstructed [?].

To summarize: the sequence of coordinate transformations

L ↦ (M, G) ↦ (dσ, dk) ↦ (dρ, dk) ↦ (h, dk) ↦ ({(f_{n}, h)}, dk)

straightens the flow of the periodic Toda lattice. From the vector

((f_{1}, h), (f_{1}, h), …, (f_{g}, h))

and the density of states dk, the operator L can be reconstructed.

10.5 The Hénon-Heils system

A famous example of a Hamiltonian systems q̇ = H_{p}, ṗ = −H_{q} is the Henon-Heils system with Hamiltonian

H(p, q) = \frac{1}{2}(p_{1}^{2} + p_{2}^{2} + q_{1}^{2} + q_{2}^{2}) + q_{1}^{2}q_{2} − \frac{1}{3}q_{2}^{2} .

It was studied first numerically by Henon and Heils in 1964 on different energy surfaces H^{−1}(E) = {p, q | H(p, q) = E = const}.

The three dimensional energy surface is compact for E ∈ [0, 1/6] and the motion stays bounded. The Poincaré section Σ = {q_{1} = 0} produces a map defined in an open subset of the plane. One measures positive topological entropy.

The Henon system is related with the Toda system with N = 3 particles. Indeed, one can show that a three particle cyclic lattice with quadratic f is equivalent to the Henon Heils system.

10.6 The integrable pendulum

The pendulum

ẍ = γ · sin x

is an integrable dynamical system. We are looking at two modifications of this integrable system, one is the the pendulum with oscillatory forcing or periodic gravitational constant, the other is and the double pendulum. Both are leading to chaotic motion. Both examples are obtained by increasing the dimension of the Hamiltonian system. The periodically forced pendulum is sometimes called a Hamiltonian system of dimension 1+1/2, the double periodic pendulum is a Hamiltonian system of dimension 2. To both systems, one can attach an area preserving map of the plane. We first perform a numerical integration of the double pendulum

RungeKuttaStep[f_,y_,y0_,dt_]:=Module[{k1,k2,k3,k4},
  k1=dt N[f /. Thread[y->y0]     ];k2=dt N[f /. Thread[y->y0+k1/2]];
  k3=dt N[f /. Thread[y->y0+k2/2]];k4=dt N[f /. Thread[y->y0+k3  ]];
  y0+(k1+2*k2+2*k3+k4)/6];
Orbit[f_,y_,y0_,t_,d_]:=Module[{q=y0,s={}},
  Do[q=RungeKuttaStep[f,y,q,d];s=Append[s,q],{Round[t/d]}];s]
g=0.25;
H=1/(1+Sin[q2]^2)*(p1^2/2 - (1+Cos[q2])*p1*p2+(p2^2/2)*(3+2 Cos[q2]))+
   g*(2(1-Cos[q1])+(1-Cos[q1+q2]));
q1dot=D[H,p1]; q2dot=D[H,p2]; p1dot=-D[H,q1]; p2dot=-Simplify[D[H,q2]];
fourtotwo[l_]:=Table[{l[[i,1]],l[[i,2]]},{i,Length[l]}];
or=Orbit[{q1dot,q2dot,p1dot, p2dot},{q1,q2,p1,p2},{0,0,1,1},10.,0.01];
Show[Graphics[{Line[fourtotwo[or]]}],Axes->Automatic]

The flow is leaving the three dimensional energy surface H = E invariant. Here, we make a Poincaré plot with the Poincaré plane q2 = 0. This gives a map on the plane (q1, p1) ↦ T (q1, p1) which is area preserving according to a Theorem of Darboux. The Hamiltonian is normalised so that we can consider the flow on the energy surface H = 1. We must be careful to check wether the initial point is in the enery surface or not.

Near[r_]:=Mod[r[[2]]+N[Pi],N[2Pi]]-N[Pi];   (* Defines Poincare surface *)
Poincare[f_,y_,y0_,dt_]:=Module[{q=y0,qq},qq=RungeKuttaStep[f,y,q,dt];
  While[(Near[q]*Near[qq] >=0.) || (Abs[Near[q]]>0.5),
  q=qq;qq=RungeKuttaStep[f,y,q,dt]];(q+qq)/2];
Orbit[y_,n_]:=Module[{q=y},s={{q[[1]],q[[3]]}};
  Do[q=Poincare[{q1dot,q2dot,p1dot, p2dot},{q1,q2,p1,p2},q,0.1];
  s=Append[s,{q[[1]],q[[3]]}],{n}];s];
pp2[qq1_,pp1_]:=Module[{s},q1=qq1;p1=pp1;q2=0.0;
  s=p2/.Solve[H==1,p2][[1]];Clear[q1,q2,p1,p2];s]

Pict[qq1_,pp1_,n_]:=ListPlot[Orbit[{qq1,0,pp1,pp2[qq1,pp1]},n]];
Pict[0.1,0.8,200];
(* Tests      *)
Poincare[{q1dot,q2dot,p1dot,p2dot},{q1,q2,p1,p2},{0.2,0.0,0.4,0.5},0.1]
RungeKuttaStep[{q1dot,q2dot,p1dot,p2dot},{q1,q2,p1,p2},%,0]

10.7 Periodic orbits for periodically driven systems

Given integers p, q ∈ N with q > 0. The space X of all maps γ : [0, q] → R, t ↦ x(t) = \frac{p}{q}t + ξ(t) with ξ ∈ W^{1,2}[0, q] satisfying ξ(t + q) = ξ(t) is a Hilbert space with the operations (ργ)(t) = \frac{p}{q}t + ρξ(t), (γ_{1} + γ_{2})(t) = \frac{p}{q}t + ξ_{1}(t) + ξ_{2}(t) and and the scalarproduct

(γ_{1}, γ_{2}) = ∫_{0}^{q} ξ_{1}(t)ξ_{2}(t) + ξ̇_{1}(t)ξ̇_{2}(t) dt .

Indeed X is the affince space p/q · t + W_{per}^{1,2}(0, q) of the Hilbert space W_{per}^{1,2}(0, q) which is a closed subspace of W^{1,2}(0, q) having periodic boundary conditions.

Given a function F ∈ C^{2}(R^{3}) satisfying F (t+1, x, p) = F (t, x+1, p) = F (t, x, p) we consider the functional

I(γ) = ∫_{0}^{q} F (t, x, ẋ)dt, γ : t ↦ x(t)

on X. One calls γ^{∗} ∈ X a minimal, if I(γ) ≥ I(γ^{∗}) for all γ ∈ X.

Lemma 10.7.1 If there exists δ > 0 such that δ ≤ F_{pp} ≤ δ^{−1}, then I is
bounded from below: μ = inf_{γ∈X} I(γ) > −∞.

Proof. From δ ≤ F_{pp} ≤ δ^{−1} follows by integration that there exists C such that

δ · p^{2}/4 − C ≤ F (T, x, p) ≤ δ^{−1}p^{2} + C .

Therefore, we have for every γ ∈ X

I(γ) = ∫_{0}^{q} F (t, x, ẋ)dt ≥ δ/4 · ∫_{0}^{q} ẋ^{2}dt − c · q ≥ cq > −∞ .

The infimum μ is finite. □

Lemma 10.7.2 The norm closure of the set

K = {γ ∈ X | I(γ) ≤ μ + 1, γ(0) ∈ [0, 1]}

is weakly compact in X.

Proof. It is enough to show the boundedness of the set K because then, the weak clousre is also bounded and so weakly compact according to the theorem of Alaoglu. Given γ ∈ K. From μ + 1 ≥ I(γ) ≥ δ/4 · ∫_{0}^{q} ẋ^{2}dt − C · q follows ∫_{0}^{q} ẋ^{2}dt ≤ 4/δ · (μ + 1) =: M_{1} and with |x(t)| ≤ x(0) + ∫_{0}^{q} ẋ(t)dt ≤ 1 + (∫_{0}^{q} ẋ^{2} · q)^{1/2} we get ∫_{0}^{q} x^{2}dt ≤ q(1 + (M_{1}q)^{1/2})^{2} =: M_{2}. Together ||γ||_{1,2} = ∫_{0}^{q} ẋ^{2} + x^{2}dt ≤ M_{1} + M_{2}. This means that K is bounded. So, also the norm closure is bounded. □

Lemma 10.7.3 a) The function p ↦ F (t, x, p) is convex:

F (t, x, p) − F (t, x, q) ≥ F_{p}(t, x, q)(p − q) .

b) If x_{n} ⇀ x, then for ϕ ∈ L^{2}[0, q] we have ∫_{0}^{q} ϕ(ẋ_{n} − ẋ) dt → 0.

c) If x_{n} ⇀ x, then for ϕ ∈ L^{1}[0, q] we have ∫_{0}^{q} ϕ(x_{n} − x)dt → 0.

Proof. a) From the condition F_{pp} ≥ δ > 0, the convexity follows automatically:

F (t, x, p) = F (t, x, q)+F_{p}(t, x, q)(p−q)+F_{pp}(t, x, p^{∗})(p−q)^{2}/2 ≥ F (t, x, q)+F_{p}(t, x, q)(p−q) .

b) The claim is clear for ϕ ∈ C^{1} by partial integration. Since C^{1} is dense in L^{2}, one can find for arbitrary ϕ ∈ L^{2} and ϵ > 0 a function ψ ∈ C^{1}, so that ||ϕ − ψ||_{2} ≤ ϵ. Then, by Schwarz inequality

| ∫_{0}^{q} ϕ(ẋ_{n} − ẋ)dt| ≤ | ∫_{0}^{q} ψ(ẋ_{n} − ẋ)dt| + 2ϵM_{1}

and consequently

lim sup_{n→∞} | ∫_{0}^{q} ϕ(ẋ_{n} − ẋ)dt| ≤ 2ϵM_{1} .

c) x_{n} →_{w} x implies that x_{n} converges uniformly to x: From ∫_{0}^{q} ẋ_{n}^{2} ≤ M_{1} follows |x_{n}(t) − x_{n}(s)| ≤ M_{1}|t − s|^{1/2} and x_{n}(t) ≤ 1 + M_{1} · \sqrt{t}. Therefore, \{x_{n}\} is a equicontinuous family of uniformly bounded functions. Arzela-Ascoli assures that a subsequence converges uniformly and because of x_{n} →_{w} x, the point x must be the limes. Therefore ||x_{n} − x||_{∞} → 0. With Hölder’s inequality follows

| ∫_{0}^{q} ϕ(x_{n} − x)dt| ≤ ∫_{0}^{q} |ϕ| dt · ||x_{n} − x||_{∞} → 0 .

□

Lemma 10.7.4 Assume that there exists ∃c > 0, so that |F_{x}| ≤ c(1 + p^{2}) and |F_{pt}| + |F_{px}| ≤ c(1 + |p|). Then, I is lower semicontinuous with respect to the weak topology: if γ_{n} ⇀ γ, then I(γ) ≤ lim inf _{n→∞} I(γ_{n}).

Proof. Given γ_{n} →_{w} γ. Then

I(γ_{n}) − I(γ) = ∫_{0}^{q} F (t, x, ẋ_{n}) + F (t, x_{n}, ẋ_{n}) − F (t, x, ẋ_{n}) − F (t, x, ẋ) dt
≥ ∫_{0}^{q} F_{x}(t, y, ẋ_{n})(x_{n} − x)dt + ∫_{0}^{q} F_{p}(t, x, ẋ)(ẋ_{n} − ẋ) dt =: D_{n}

and y(t) is in the interval [x_{n}(t), x(t)]. For the inequality, we have used Lemma 3a). Since F_{x} is in L^{1}[0, q] (because |F_{x}(x, ẋ, t)| ≤ c(1 + ẋ^{2})) and F_{p} is in L^{2}[0, q] (|F_{pt}| ≤ c(1 + |p|) implies |F_{p}| ≤ c_{1}(1 + |p|)), we conclude with Lemma 3b)c), that D_{n} → 0 for n → ∞. This finishes the proof: lim inf_{n→∞} I(γ_{n}) − I(γ) ≥ 0. □

Theorem 10.7.5 A minimal γ^{∗} of I exists. and for every ϕ ∈ X, the first variation lim_{ϵ→0}ϵ^{−1}(I(γ^{∗} + ϵϕ) − I(γ^{∗})) exists and is vanishing.

Proof. A lower semicontinuous function which is bounded from above takes its minimum on a compact space. Let γ^{∗} : t ↦ x^{∗}(t) be the minimal element in X from which we have shown the existence. Then I(γ^{∗} + ϵϕ) ≥ I(γ^{∗}) for all ϕ ∈ X , ϕ : t ↦ y(t). This means that the first variation is vanishing if it exists. We have

ϵ^{−1}(I(γ^{∗} + ϵϕ) − I(γ^{∗})) = ϵ^{−1}(∫_{0}^{q} F (t, x^{∗} + ϵy, ẋ^{∗} + ϵẏ) − F (t, x^{∗}, ẋ^{∗})dt)
= ∫_{0}^{q} λ(t, ϵ)ẏ + μ(t, ϵ)ydt ,

where

λ(t, ϵ) = ∫_{0}^{q} F_{p}(t, x^{∗} + θϵy, ẋ^{∗} + θϵẏ)dθ

and

μ(t, ϵ) = ∫_{0}^{q} F_{x}(t, x^{∗} + θϵy, ẋ^{∗} + θϵẏ)dθ .

This can be estimated as follows:for ϵ < 1 and θ_{0} ∈ [0, 1]

|λ(t, ϵ)| ≤ c(1 + |ẋ^{∗} + ϵθ_{0}ẏ|) ≤ c(1 + |ẋ^{∗}| + |ẏ|)
|μ(t, ϵ)| ≤ c(1 + (ẋ^{∗})^{2} + ẏ^{2}) .

According to Lebesque convergence theorem, the functions λ(t, ϵ)ẏ and μ(t, ϵ)y are in L^{1}[0, q], since the majorants c(1 + |ẋ^{∗}| + |ẏ|)ẏ and c(1 + (ẋ^{∗})^{2} + ẏ^{2}) are Lebesque integrable. the convergence lim_{ϵ→0}((I(γ^{∗} + ϵϕ) − I(γ^{∗}))/ϵ follows. □

Lemma 10.7.6 The integrated Euler equations hold for γ^{∗} : t ↦ x^{∗}: There exists C ∈ R such that F_{p}(t, x^{∗}, ẋ^{∗}) = ∫_{0}^{t} F_{x}(s, x^{∗}, ẋ^{∗})ds + C.

Proof. We have

lim_{ϵ→0}((I(γ^{∗} + ϵϕ) − I(γ^{∗}))/ϵ = ∫_{0}^{q} F_{p}(t, x^{∗}, ẋ^{∗})ẏ + F_{x}(t, x^{∗}, ẋ^{∗})y dt =
= ∫_{0}^{q}[F_{p}(t, x^{∗}, ẋ^{∗}) − ∫_{0}^{q} F_{x}(s, x, ẋ)ds + C]ẏ dt = 0

which implies

F_{p}(t, x^{∗}, ẋ^{∗}) = ∫_{0}^{t} F_{x}(s, x, ẋ) ds + C .
□

Corollary 10.7.7 a) t ↦ F_{p}(t, x^{∗}(t), ẋ^{∗}(t)) is absolutely continuous.

b) t ↦ ẋ^{∗}(t) is continuous.

c) t ↦ F_{p}(t, x^{∗}(t), ẋ^{∗}(t)) is continuously differentiable

d) t ↦ ẋ^{∗}(t) is continuously differentiable.

e) t ↦ x^{∗}(t) is twice differentiable (it is regular).

Proof.

a) From the last lemma follows that F_{p}(t, x^{∗}, ẋ^{∗}) = ∫_{0}^{t} F_{x}(s, x, ẋ)ds + C and that F_{p} is absolutely continuous.

b) From F_{pp} > 0 and the implicit function theorem, we get ẋ^{∗} ∈ C^{0} and so

c) x^{∗} ∈ C^{1}.

d) Using the integrated Euler equations we have F_{p} ∈ C^{1}. Again with the implicit function theorem we have ẋ^{∗} ∈ C^{1}.

e) Therefore x^{∗} ∈ C^{2}[0, q]. □

Corollary 10.7.8 The minimal is regular and satisfies the Euler equation \frac{d}{dt}F_{p}(t, x^{∗}, ẋ^{∗}) = F_{x}(t, x^{∗}, ẋ^{∗})

Proof. Differentiate the integrated Euler equations. This is possible since x^{∗} ∈ C^{2}[0, q]. □

We have shown the following classical theorem in calculus of variations:

Theorem 10.7.9 Assume F ∈ C^{2}(R^{3}) satisfies ∃δ > 0, ∃c > 0 δ ≤ F_{pp} ≤ δ^{−1}, |F_{x}| ≤ c(1 + p^{2}) and |F_{pt}| + |F_{px}| ≤ c(1 + |p|). Then, the functional I(γ) = ∫_{0}^{q} F (t, x, ẋ)dt, γ : t ↦ x(t) on the Hilbert space X has a critical point γ^{∗} which is a minimum is in C^{2}[0, q] and satisfies the Euler equations.

\frac{d}{dt}F_{p}(t, x^{∗}, ẋ^{∗}) = F_{x}(t, x^{∗}, ẋ^{∗}).

This gives the following application to the nonlinear pendulum:

Corollary 10.7.10 Given a potential V ∈ C^{2}(R^{2}) defined by V (t, x) = g(t)cos(2πx), where g(t + 1) = g(t) and F (t, x, p) = p^{2}/2 + V (t, x). One can think of g(t) as a periodically (for example by the tidal forces) perturbed gravitational field. For every p, q > 0, there exist periodic orbits such that the pendulum makes in time q exacly p turns.

Proof. Check the conditions F ∈ C^{2}, F_{pp} = 1 and F_{x} ≤ const and F_{pt} = F_{px} = 0. □

10.8 Epilog

Newtonian systems The Newtonian many body problem is probably the oldest problem in mechanics. The main motivation to study such systems is of course the solar system. Still unsolved is the problem whether the solar system is stable. (See the article [?], where the stability problem of the solar system is motivating the stability problem of the exteriour billiard. Also unknown is whether the set of initial configurations in the 6N dimensional phase space of the Newtonian N body problem which have a global solution, has full Lebesgue measure. This is one of Simons 15 problems in mathematical physics. (problem 1A in [?]). There is also no example a set of positive measure in the phase space of a Newtonian or N body problem where one has positive metric entropy.

Vortex systems The motion of finitely many vortices is an interesting Hamiltonian system showing integrable and chaotic motion. It is an example of a Hamiltonian system which is not derived by the Legendre transformation of a Lagrange system. Point vortices exist in superfluid helium. Vortex motion can simulate the Euler equations of a two-dimensional incompressable fluid. They discribe singular solutions of the two dimensional Euler equations. The chaotic motion of vortices should shed some light to turbulence in fluid dynamics. The question, what is the measure of the set of initial conditions leading to a collaps in R^{2} seems to be unsolved. (see Dürr). It is also unknown if for an arbitrary domain the motion of two vortices is already chaotic.

Some history of the vortex problem: Vortex theory was introduced by Helmholtz one century ago. Kirchoff and Poincaré outlined the Hamiltonian structure of the system. Kirchhoff 1876 invents the model of finitely many vortices. Often the system is also called Kirchhoff problem. He remarks that the equations of a point-vortex motion in an unbounded plane defines a Hamiltonian system. Lin (Caltech) 1943 investigates the motion of vortices in two dimensions in arbitrary domain. There is much numerical work on finitely many vortices in the plane giving numerical evidence that the motion of four vortices is chaotic. Ziglin 1980 argues with a Melnikov perturbation argument that the problem of four vortices has a horse shoe and is therefore not integrable. Marsden, Weinstein remark 1983 that the proof of Ziglin has some gaps and some discussion followed. For an other 4 vortex distribution [?] show the nonintegrability. Recent work [?] give new chaotic regions. The statistical mechanics of a gas of vortices in the thermodynamic limit has been studied by Ruelle and Fröhlich. Using KAM theory Khanin showed that in the phase speac of any system with an arbitrary number of ovortices,s there exists a set of initial conditions of positive measure for which the motion of vortices is quasi-periodic. Celletti and Falcolini [?] have (using computer assisted proofs) quantitative results for the break-down of invariant tori in the four vortex problem.

Toda systems The study of nonlinear lattices started seriously in the 50’s, when Fermi,Pasta and Ulam studied the problem of nonlinear lattices

q̈_{n} = f(q_{n+1} − q_{n}) − f(q_{n} − q_{n−1}) .

These experiments were amoung the first computer experiments. Fermi,Pasta and Ulam experimented with quadratic, cubic and piecewise linear functions f and N = 32 or N = 64 particles. One expected at that time that such a system would evolve chaotically and was surprised that recurrence phenomena occured. It was still more surprising that one would get an integrable system when f is an exponential function which gives the Toda system. The recurrence discovery for the quadratic, cubic lattice can be explained partly, that one is not too far from an integrable sytem. Toda systems have been discovered by Toda in 1967. There exist many versions of such systems. In general, it is an isospectral deformation of Jacobi matrices. The integration of such systems involve differend kind of mathematics like scattering theory algebraic geometry or the spectral theory of discrete Schroedinger operators.

10.9 Exercices

Exercice 10 (last exercice)

Topic: Equilibria in the restricted three body problem

We are considering the restricted three body problem

ẍ = 2ẏ + \frac{∂V}{∂x}
ÿ = −2ẋ + \frac{∂V}{∂y}

with

V (x, y) = \frac{x^{2} + y^{2}}{2} + \frac{1 − μ}{r_{2}(q)} + \frac{μ}{r_{1}(q)}

and r_{1}^{2} = (q_{1} − 1 + μ)^{2} + q_{2}^{2}, r_{2}^{2} = (r_{1} + μ)^{2} + q_{2}^{2}.

A relative equlibrum of the restricted three body problem is a stationary solution (x(t), y(t)) = (x(0), y(0)) of the above differential equations.

The coordinates (x(t), y(t)) describe the motion of an asteroid for example Achilles in the field of the sun S = (−μ, 1) and Jupiter J = (1 − μ, 0) where the other planets and the disturbence of the asteriod are neglected.

The asteroid Achilles was discoverd 1907 and moves practically on Jupiter’s orbit being constantly ahead of it. Five other asteroids, the Trojans were also discoverd on the oposite side.

1) Show that there are 5 equilibrium solutions of the restricted three body problem. They can be devided into 2 equilateral libration points and 3 collinear libration points.

2) (Lagrange) Under which conditions at μ are the two equilateral libration points elliptic?

3) (*) Show that for the real Sun-Jupiter system, the equilateral libration points are elliptic. (this is a first explanation why the Trojans and Archilles can be found, where they are. One can actually show that all equilateral libration points are stable but this is beyond the scope of simple arguments). Datas: The mass 1−μ of the sun is about 329 390 times the mass of the earth. The mass μ of Juiter is about 318 times the mass of the earth.

4) (*) Show that the collinear libration points are unstable. This question might be easier to answer with Mathematica or Maple.

Dynamical systems in number theory

11.1 The Collatz problem

Consider the following dynamical system on N:

T : N → N is defined by T (n) = n/2 if n is even and T (n) = 3n + 1 if n is odd.

The ”3n + 1 Conjecture is: The periodic orbit 1 ↦ 4 ↦ 2 ↦ 1 attracts the whole set N.

A Mathematica program for investiagation of the map is very simple. The following procedures compute the time, it needs for a natural number to reach the cycle 1 ↦ 4 ↦ 2 ↦ 1.

T[n_]:=If[OddQ[n],3n+1,n/2];
L[n_]:=Module[{i=1, m=n}, While[m != 1, m = T[m]; i++];i];
ListPlot[Table[L[n],{n,5000}]]

The same definition of the map can be applied to a map on Z. There is the following finite cycle conjecture: there are only finitly many periodic orbits of T on Z. Examples are the fixed points 0, 1, −1, periodic orbits of length 3 starting at −5 or length 11 starting at n = −17. An other stronger conjecture says that these periodic orbits are the only ones. It is also conjectured that there is no unbounded orbit on Z converging to infinity.

The map T : Z → Z can be extended to a map on the space of dyadic integers

G = {α = (a_{1}, a_{2}, …,), a_{i} ∈ {0, 1}}

which is a compact abelian topological group with group operation

(ω + η)_{n} = ω_{n} + η_{n} + ρ_{n−1} (mod 2) ,

where ρ_{0} = 0 and ρ_{n} ∈ {0, 1} is equal to 1 if and only if ω_{n} + η_{n} + ρ_{n−1} ≥ 2. The inverse is formed by

a^{−1} = \overline{a} + 1

where \overline{a} is the sequence obtained by exchanging the alphabet 1 ↔ 0 and 1 = (1, 0, 0, …) is the one-element. (The group G is also a ring).

This map which we call Collatz map is measure preserving with respect to the product measure (Haar measure) and is strongly mixing. This says nothing about the behavior on the subset

N = {n · 1 | n ∈ N}

which has measure zero. We describe now the Collatz map on the group G of dyadic integers. Denote by

σ_{0} : a = (a_{1}, a_{2}, …) → (0, a_{1}, a_{2}, …) .

one branch of the inverse of the shift

σ : a = (a_{1}, a_{2}, …) → (a_{2}, a_{3}, …) .

Call G_{0} = σ_{0}G and G_{1} = G \ σ_{0}G. Define T : G → G as

T (x) = \{ \begin{matrix}σx & , x ∈ G_{0} \\ σ_{0}(x) + x + 1 & , x ∈ G_{1} \end{matrix} .

The measure μ on G is the same as the the Bernoulli measure on G = {0, 1}^{N}.

Remark. The map T is not invertible. For example both (1, 0, 1, 0, 0, 0) and (0, 0, 0, 0, 0, 1, 0, …) go to (0, 0, 0, 0, 1, 0, …), this paraphrases 5, 32 ↦ 16.

The Collatz problem is in the language of dynamical systems: What is the basin of attraction of the periodic orbit

1 ↦ (0, 0, 1) ↦ (0, 1) ↦ 1 ?

Is every element in N = {n · 1 n ∈ N} in the basin of attraction? The general conjecture is that this should be true.

If the conjecture is true, we know that the map is on the countable subset N not so interesting. Everthing is attracted to a periodic orbit of period 3. From the measure theoretical point of view, the map is conservative:

A measurable map is measure preserving, if m(T^{−1}(Y)) = m(Y) for all measurable sets Y. It is ergodic if m(Y)^{2} = m(Y) for all measurable sets. It is mixing, if m(T^{−n}(Y) ∩ Z) → m(Y)m(Z) for all measurable sets Y, Z.

Proposition 11.1.1 (Matthews,Watts)

The Collatz map T is measure preserving and mixing.

Proof. We show only that the map T is measure preserving. In order to see that T is measure preserving with respect to the Haar measure, it is enough to show that the cylinder sets

G_{a}(n) = {x ∈ G | x_{n} = a}

satisfy m(T^{−1}G_{a}(n)) = m(G_{a}(n)). Because this is true for the map T : G_{0} ↦ T G_{0} We have only to show that it is also true for the map T : G_{1} → G_{0}, x ↦ σ_{0}x + x + 1. This means that we have to show that the map T̃ : G_{1} → G_{1}, x ↦ σ_{0}x + x is measure preserving. We compute the inverse of a cylinder set

G_{0}(n) = {y | y_{n} = 0}

with n ≥ 2. (For the cylinder set G_{1}(1) = G_{1}, we have T̃(G_{1}) = G_{1}.) The set T̃^{−1}(G_{0}(n)) is

{x | (σ_{0}x + x)_{n} = 0} = {x_{n−2} = 0, x_{n−1} = 0, x_{n} = 0}
∪ {x_{n−2} = 1, x_{n−1} = 0, x_{n} = 0}
∪ {x_{n−2} = 0, x_{n−1} = 1, x_{n} = 1}
∪ {x_{n−2} = 1, x_{n−1} = 1, x_{n} = 0} .

The right hand side is a disjoint union of sets having together measure 1/2 which is also the measure of the cylinder set G_{0}(n). □

Citations about this problem: Erdös : ”Mathematics is not yet ready for this problem.” Kakutani: ”This problem is part of a conspiracy to slow down mathematical research in the US”.

Prizes for solving the problem: Coxeter $ 50, Erdös $ 500, Thwaites £ 1000.

Related problems. If one replaces 3n + 1 by 7n + 1 one expects divergent trajectories. The starting value of 3 for example is conjectured to diverge If we run this

T[n_]:=If[OddQ[n],7n+1,n/2];
L[n_]:=Module[{i=1,m=n},
  While[m != 1,m=T[m];If[Mod[i,1000]==0,Print[N[Log[10,m]]]];i++];i];
L[3]

we number of digits grow in each 1000 steps for about 80 digits. A run of about 1/2 hour gave digits with 12000 digits. It seems that the numbers grow exponentially.

11.2 Pseudo random generators

It is not so long ago that the question ”what is randomness” has not even made sense. Today, having an axiomatic mathematical setting of probability theory, one can work with randomness quite well. Often, a problem is only present, because the right setting has not been achieved. The mathematical setting of playing dices is as follows. Define the probability space {0, 1}^{Z} equiped with the product measure m. The shift map (T x)_{n} = x_{n+1} is measure preserving. The dynamical system (X, T, m) is called a Bernoulli shift. The random variable d(x) = x_{0} is the outcome of an experiment. The event d^{−1}({1} is a subset called ”tail”, the event d^{−1}({0}) is called ”head”. The sequence d(T^{n}x) is the outcome of an experiment by throwing dices.

A ”random” {0, 1} sequence should have all the properties which holds for almost all sequences. Examples of such properties are: The ”law of large numbers”:

\frac{1}{n} ∑_{n=0}^{∞} d(T^{n}x) → 1/2 .

Normality: for all N, the words

(d(x), d(T x), d(T^{2}x), …, d(T^{N}x))

are uniformly distributed. (all blocks of length k occur with the same frequency.)

One can not define properly, what is a random sequence. One can therefore also not tell from a specific sequence, if it is random because the definition of random sequence is not done. However, one can fix finitely or countably many properties and look at the set of sequences, which have this property.

It is very hard and probably impossible in general to decide wether a given sequence has some property like normality. It is for example unknown, if the sequence x_{i} defined by

π = ∑_{i=−1}^{∞} x_{i}2^{−i}

is normal.

In computers, random sequences are usually generated by pseudo random generators, which are deterministic dynamical systems. One needs random numbers in many so called Monte-Carlo algorithms, private key cryptosystsm. An advantage of pseudo random numbers is that they are reproducible. It is a difficult business to find ”good” random generators. It is even difficult to find out, what is a ”good” random generator which means to find a sequence of tests for randomness. There are books filled with millions of random digits, physical processes like ”varactor diodes” to produce random bits. There is a variety of good pseudo random generators. We are going to look at some of them from the point of view of dynamical systems.

Let X = (Z_{N} = Z/(NZ), +, ·) be the ring of numbers modulo N. We are going to give maps T : Z_{N} → Z_{N} which are used as pseudo random generators.

• Congruential generator: a, b parameters

T (x) = ax + b .

• Power generator: d parameter

T (x) = x^{d} .

• The quadratic map: c parameter

T (x) = x^{2} + c .

• Exponential generator: g parameter

T (x) = g^{x} .

• Kneeding map in X × X

T (x, y) = (y + f(x), x) .

• Shift register in X^{d}

(x_{1}, …, x_{d}) ↦ (f(x_{1}, … x_{d}), x_{1}, … x_{d−1})

One can say, that this is a recursion x_{n} = f(x_{n−1}, x_{n−d}).

• Cellular automata in X^{d}

(x_{1}, …, x_{d}) ↦ (f(x_{d}, x_{1}, x_{2}), f(x_{1}, x_{2}, x_{3}), … f(x_{d−1}, x_{d}, x_{1})) .

In principle, all these examples have analogues for dynamical systems where X is replaced by R^{n} or C^{n}. A computer simulation of such a systems is always a map on a finite set and every orbit is eventually periodic. It is however not trivial to understand the roundoff-arithmetic and to find a good model of this finite map. The projection of the dynamical system to a finite permutation is also depending on the machine and the algorithm.

11.3 Encryption algorithms

The data encryption standard (DES) is a famous secret-key cryptosystem set forward in 1977 by the National Bureau of Standards. It was designed to be used for a period of ten to fifteen years ”within the Federal Bovernment for the cryptographic protection of sensitive but unclassified computer data”. (crf. [?] p. 14).

We will only roughly go through the encryption algorithm as a dynamical system. The precise algorithm can be found in [?]. Let

X = Y × Y = {0, 1}^{32} × {0, 1}^{32} = {{0, 1}^{64}

be the set of all 64bit data blocks. We write a point z ∈ X as a pair z = (x, y) of a ”two dimensional plane”, where x, y are in Y = {0, 1}^{32}. Let K be the parameter set K = {0, 1}^{56}. The points c in K are called keys. For each c ∈ K, there is a defined map f_{c} : Y ↦ Y. Given is also a permutation S of X. Denote with U_{c} the ”monotone twist map”

(x, y) ↦ (y + f_{c}(x), x) ,

where + is the exclusiv-or operation. The DES dynamical system is given by

T_{c}(z) = S^{−1}U_{c}^{16}S(z) .

The map T_{c} is just a conjugation to an 16’th iteration of a monotone twist map.

Actually the story is a little bit more complicated in that in the real DES encryption in each round an other key c_{i} ∈ {0, 1}^{48} is computed from c. We have then acutally

T_{c}(z) = S^{−1}U_{c_{1}} … U_{c_{16}}S(z)

where U_{c_{i}} are twist maps.

We illustrate with the Standard map how the result is sensitive on initial conditions. In the following mathematica procedures, one forces the program to calculate with a specific accuracy N. We make 16 rounds like in the DES encoding.

SP[x_,K_]:=SetPrecision[x,K]
T[{x_,y_},g_,K_]:=SP[N[Mod[{x+y+g*Sin[x],y+g*Sin[x]},SP[2 Pi,K]]],K];
Orbit[p0_,n_,g_,K_]:=Module[{p=p0},Do[p=T[p,g,K],{n}];p];
R[K_]:=Orbit[{SP[0.1,K],SP[3.4,K]},16,SP[20.0,K],K];

The values of R[6] and R[7] are completetly different.

11.4 The discrete Logarithm problem

Given a dynamical system (X, T). The Logarithm problem is the problem to find for a given pair of points x, y ∈ X a number n ∈ Z such that

T^{n}x = y

or to tell that no such number exists. The classical discrete logarithm problem (used for example for secrete key exchanges like the Diffie-Helleman scheme) is obtained with the dynamical system (Z_{p}^{∗}, T)

T : x ↦ a · x mod p ,

where Z_{p}^{∗} = {1, 2, … p − 1} is the multiplicative group modulo p and p is prime. The discrete logarithm problem is the problem to find the number n in the equation

a^{n} = b mod N .

In this context there is a still standing challenge called ”the 100 dollar problem” of McCurley (1990). Given a prime number p. Known are a1 = 7^{x}(modp), a2 = 7^{y}(modp), where x, y are not known. McCurley will pay 100 to the first person who 7^{xy}(modp) and proof that this is so.

q=(Power[7,149]-1)/6;
p=2*739*q+1;
a1=127402180119973946824269244334322849749382042586931621\
   654557735290322914679095998681860978813046595166455458\
   144280588076766033781;
a2=1801622852874531024447828348367998501596704669534669731\
   3025121734059953772058475958176910625380692101651848662\
   362137934026803049;

11.5 Recurrence and the theorem of van der Waerden

Introduction. Given an arbitrary partition of the entire numbers Z = A_{1} ∪ A_{2} ⋯ A_{q}. Does there exists an A_{i}, in which there exists arbitrary long arithmetic progressions ? In 1927, B.L van der Waerden solved this so called Baudet conjecture which had occupied for a long time a big part of the mathematical society of that time. His proof [?] was completely combinatorial and not easy. A considerably simpler proof was found by the Russian mathematician M.A. Lukomskaja. Her still elementary proof was published in the beautyful little book of Chinchin ??. Furstenberg ?? gave a very simple proof for van der Waerden’s theorem by reformulating the question as a problem for multi-dimensional dynamical systems (X, T_{1}, T_{2}, …, T_{d}, m) where T_{i} are continuous commuting maps on the compact metric space X. A point x ∈ X is called multiple recurrent if there exists a sequence n_{k} → ∞ such that T_{i}^{n_{k}}x → x for all i = 1, … l.

Theorem 11.5.1 (Multiple Birkhoff Recurrence theorem) There exists a multiple recurrent point in X.

A proof can be found in Fuerstenbergs book. Instead we will show how the theorem of van der Waerden follows from it.

Theorem 11.5.2 (Van der Waerden) Given a finite partition of the entire numbers Z = A_{1} ∪ A_{2} ⋯ A_{q}. There exists a set A_{i}, in which there exists arbitrary long arithmetic progressions.

Proof. It is enough to show that for each l ≥ 1 there exists an A_{j} so that A_{j} contains an arithmetic progression of lenght l. Some A_{j} will occur then infinitly often and this will be our desired A_{j}.

Define Λ = {1, ⋯ q}. Look at the compact metric space X = Λ^{Z} with metric

d(x, y) = 2^{− inf_{x_{i}≠y_{i}} |i|} .

The shift transformation x_{i} ↦ T x_{i+1} is a homoeomorphism of X. There is a bijection between elements in X and partitions of Z in q sets A_{1}, ⋯ A_{q}: n ∈ A_{j} ⇔ x_{n} = j ∈ Λ. In order to show the theorem of van der Waerden, it is enough to show for a given x ∈ X the assertion

∀l ≥ 1 ∃m ∃n ≥ 1 x_{m} = x_{h+n} = x_{h+2n} = ⋯ x_{h+nl}

holds. The reason is that we have found then an arithmetic progression of lenght l in one of the sets A_{j}. This statement follows immediatly from Birkhoff’s ergodic theorem because if d(T^{i}x, T^{j}x) < 1 for x ∈ X, then x_{i} = x_{j}. □

It is possible to prove also a multidimensional version of the theorem of van der Waerden, which was found by Gruenwald. The proof presented here is also from Furstenberg:

Theorem 11.5.3 (Gruenwald) Given a finite partition A_{1}, A_{2}, ⋯ A_{q} of G = Z^{n}. There exists a set A_{j} such that for each finite set F ⊂ G there exists a ∈ G and b ∈ Z such that a + bF ⊂ B_{j}.

Proof. Write F = {f_{1}, … f_{l}}. Take as before the set Λ = {1, ⋯ q} and form the compact metric space Ω = Λ^{G}. As before we can find a bijective corespondence between Ω and partitions of G with q elements. Define the transformations T_{i} : w_{n} ↦ w_{n+f_{i}}. Given a point w ∈ Ω. We denote with X the smallest closed set in Ω which contains w and is invariant under the transformations T_{i}. The multiple Birkhoff recurrence theorem gives a point x ∈ X and b ∈ Z such that d(T_{i}^{b}x, x) < 1 for i = 1, ⋯ l. This means, that x_{0} = x_{b·e_{i}} and we have therefore an a ∈ G such that w_{a} = w_{a+b_{i}}. □

It is in general very difficult to decide wither a sequence of integers contains arbitrary long arithmetic progressions. Here is a problem of Erdoes who promises 3000 dollars for its solution:

Let a_{1} < a_{2} < … be a sequence of integers such that ∑ a_{n}^{−1} is diverging. Then the a_{n}’s contain arbitrariliy long arithmetic progressions. (See [?].)

Additional exercices

Endterm MA 91, MA 191 k

5 hours time limit. All literature allowed. Due: March 17, 20^{00} in my mailbox. The difficulties of the questions varies.

Arnold’s cat map and Sinai’s family

1) Given Arnold’s cat map

T (x, y) = (2x + y, x + y) mod1

on the torus T^{2}. Show that T is bijective and preserving the volume measure.

2) Show that the periodic orbits are dense on the torus and that every periodic orbit is hyperbolic.

3) Calculate the Lyapunov exponent

λ(x) = lim_{n→∞} \frac{1}{n} log(||dT^{n}(x)||

for all x ∈ T^{2}.

4) Assume, we have two points x, y on the torus T^{2} with distance 10^{−k}. After how many iterations of T does one expect an error of order 1?

5) Show that T is chaotic in the sense of Devaney. Hint: It is useful to look at the map T defined on R^{2} and look at the eigenspaces of a fixed point.

6) Show that the Sinai family

T_{γ}(x, y) = (2x + y + γ sin(x + y), x + y)

with real parameter γ is measure preserving and that it is topologically conjugated to T_{0}(γ) if |γ| is small enough. Hint: look at the theorem of Grobmann Hartmann.

7) Conclude that that for |γ| small enough, the Sinai family is chaotic in the sense of Devaney.

Nowhere contracting maps

8) Let (X, d) be a compact metric space and let T : X → X be a continuous map.

9) Definition: A point x ∈ X is called recurrent, if there exists for all ϵ > 0 a number n ∈ N, such that d(T^{n}(x), x) < ϵ. Definition: A map T is called nowhere contracting if d(T x, T y) ≥ d(x, y) for all x, y ∈ X.

10) Assume T is nowhere contracting. Show that the map is injective and that any point x ∈ X is recurrent.

Equip the space Z = X × X with the metric

δ((x_{1}, y_{1}), (x_{2}, y_{2})) = d(x_{1}, x_{2}) + d(y_{1}, y_{2})

and define the map S(x, y) = (T x, T x).

11) Show that Z is a compact metric space and that S is a continuous map on Z.

12) Show that

f : Z = X × X → R, z = (x, y) ↦ d(x, y)

is continuous.

13) Show that S = T × T is nowhere contracting on Z = X × X if and only if T is nowhere contracting on X.

Definition: A continuous map is an isometry if

d(T x, T y) = d(x, y), ∀x, y ∈ X .

14) Prove the following theorem:

Theorem: Given a compact metric space (X, d) and a continuous map T : X → X which is nowhere contracting. Then, the map is an isometry.

(Hint: use the already proven information).

15) Which of the ingredients of this theorem are necessary: the continuity of T, the compactness of X? Give a counterexample of the theorem for the assumption(s) necessary in the theorem.

16) Assume T is nowhere contracting. Show that T_{H} : A ↦ T (A) on the compact metric space (C, d_{H}) with the Hausdorff metric d_{H} is an isometry.

17) Show that dim_{H}(T (A)) = dim_{H}(A) for all A ∈ C if T is nowhere contracting.

Zaslavsky’s map

18) Define the two maps

R_{α} : (x, y) ↦ (cos(α)x + sin(α)y, − sin(α)x + cos(α)y)
L_{γ} : (x, y) ↦ (x + γ · sin(y), y)

in the plane R^{2}. The rotation angle α and the coupling constant γ are real parameters. The map T_{α,γ} = R_{α}L_{γ} is called Zaslavsky’s map.

19) Show that T_{α,γ} is volume preserving for all parameters.

20) Find parameters for which the map T_{α,γ} has a shift embedded.

21) For which parameters can the map be defined as a map on the torus T^{2}?

22) Find a parameter α, for which the map T_{α,γ} has a dense set of periodic orbits.

Exterior billiards

23) Let C be the set of all compact sets in R^{2}. Denote with K ∈ C is convex, if every intersection of K with a line is a connected set (an interval, a point or the empty set). Define K ⊂ C as the set of convex subsets in R^{2}. Show that K is a compact subset of C.

24) Given K ∈ K. Denote with T_{K} the exterior billiard map at the boundary of K. Let S : R^{2} ↦ R^{2} is a linear invertible map. Show that S leaves invariant the subset K of C.

25) Show that the billiard map T_{K} is isomorphic to the billiard map T_{S(K)} if S is a linear invertible map.

26) Analyse the exterior billiard map in the case, when

K = {3^{2}x^{2} + 5^{2}y^{2} ≤ 1} .

27) Analyse the exterior billiard map in the case, when

K = {y = 0, 0 ≤ x ≤ 1} .

Interior billiards

28) Show that the interiour billiard maps at a circle and an ellipse are in general not isomorphic. (Hint: Look at periodic orbits).

29) A billiard map can be defined on any compact two dimensional manifold M with smooth boundary. The ball moves in the interior of the surface on geodesics and is reflected at the boundary with the usual reflection law. Let M be the half sphere

M = {(x, y, z) | x^{2} + y^{2} + z^{2} = 1, z ≥ 0}

which has the boundary Γ = {(x, y, x) | x^{2} + y^{2} = 1, z = 0}. The billiard ball moves on the half sphere on great circles until it hits the boundary, is reflected and moves again on great circles. The return map to the boundary defines a billiard map. Describe coordinates for this billiard map on the half sphere, in which the map is area-preserving.

30) Show that the billiard map on the manifold

M = {(x, y, z) | x^{2} + y^{2} = 1, z ≥ 0, x + z ≤ 2}

is isomorphic to the billiard map on some convex curve in R^{2}. Give the geometry of this convex curve.

31) Find a manifold of the form

M = {(x, y, z) | x^{2} + y^{2} = 1, z ≥ 0, F (x, z) ≤ 0}

which gives a billiard map isomorphic to Bunimovich’s stadion.

32) Show that the billiard in the manifold

M = {(x, y, z) | x^{2} + y^{2} = 1, 2π ≥ z ≥ 0, }

has a dense set of periodic orbits but that the map is not chaotic in the sense of Devaney.

Bibliography

[1] R. Ainsley. Bluff your way in maths. Ravette, London, 1988.

[2] J. Banks, J.Brooks, G.Cairns, G.Davis, and P.Stacey. On Devaney’s definition of chaos. Amer. Math. Monthly, 99:332–334, 1992.

[3] M. Barnsley. Fractals everywhere. Academic Press, Boston, 1993.

[4] A. Beardon. Iteration of rational functions. Graduate Texts in Mathematics. Springer-Verlag, New York, 1990.

[5] M. Benedicks and L-S. Young. Sinai-bowen-ruelle measures for certain Hénon maps. Inv. Math., 112:541–576, 1993.

[6] P. Bergé, Y.Pomeau, and C.Vidal. Order within Chaos. John Wiley and sons, 1984.

[7] G.D. Birkhoff. On the periodic motions of dynamical systems. Acta Math., 50:359–379, 1950.

[8] P. Blanchard. complex analytic dynamics on the riemann sphere. BAMS, 11:85–141, 1984.

[9] B. Branner. The mandelbrot set. volume 39 of Proceedings of Symposia in Applied Mathematics, pages 75–105. American Mathematical Society, 1989.

[10] L.A. Bunimovich and Ya. G. Sinai. Spacetime chaos in coupled map lattices. Nonlinearity, 1:491–516, 1988.

[11] L. Carleson and T.W. Gamelin. Complex Dynamics. Springer-Verlag, New York, 1993.

[12] J.H. Curry. On the Hénon transformation. Commun. Math. Phys., 68:129–140, 1979.

[13] W. de Melo and S. van Strien. ONe dimensional dynamics, volume 25 of Series of modern surveys in mathematics. Springer Verlag, 1993.

[14] R. Devaney. Triple collisions in the planar isosceles three body problem. Inv. Math., 60:249–267, 1980.

[15] R. Devaney and Z. Nitecki. Shift automorphisms in the Hénon mapping. Commun. Math. Phys., 67:137–146, 1976.

[16] R.L. Devaney. An introduction to Chaotic dynamical systems. The Advanced Book Program. Addison-Wesley Publisher Company, Inc, Redwood City, second edition, 1989.

[17] R.L. Devaney. A first course in Chaotic dynamical systems, theory and experiments. The Advanced Book Program. Addison-Wesley Publisher Company, Inc, Redwood City, second edition, 1992.

[18] V. Donnay. Using integrability to produce chaos: billiards with positive entropy. Commun. Math. Phys., 141:225–257, 1991.

[19] G.A. Edgar. Measure,Topology and fractal geometry. Undergraduate Texts in Mathematics. Springer-Verlag, 1990.

[20] I. Ekeland. Mathematics and the unexpected. The university of Chicago press, 1988.

[21] A. Eremenko and M. Lyubich. The dynamics of analytic transformations. Leningrad Math. J., 3:563–634, 1990.

[22] K.J. Falconer. The geometry of fractal sets. Cambridge University Press, Cambridge, 1985.

[23] M. Field and M. Golubitsky. Symmetry in chaos, a search for attern in mathematics, art and nature. Oxford University press, Oxford, 1992.

[24] D. Gulick. Encounters with chaos. McGraw-Hill, New York, 1992.

[25] V.M. Gundlach and D.A. Rand. Spatio-temporal chaos: 1. Hyperbolicity, structural stability, spatio-temporal shadowing and symbolic dynamics. Nonlinearity, 6:165–200, 1993.

[26] V.M. Gundlach and D.A. Rand. Spatio-temporal chaos: 2. Unique Gibbs states for higher dimensional symbolic systems. Nonlinearity, 6:201–213, 1993.

[27] V.M. Gundlach and D.A. Rand. Spatio-temporal chaos: 3. Natural spatio-temporal measures for coupled circle map lattices. Nonlinearity, 6:215–230, 1993.

[28] B. Hasslacher. Discrete fluids. In N. Grant Cooper, editor, From cardinals to chaos: reflections on the life and legacy of Stanislaw Ulam, pages 175–217. Cambridge University Press, 1989.

[29] G.A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theor., 3:320–375, 1969.

[30] M. Hénon. A two-dimensional mapping with a strange attractor. Commun. Math. Phys., 50:69–77, 1976.

[31] A. Hof and O. Knill. Cellular automata with almost periodic initial conditions. Nonlinearity, 8:477–491, 1995.

[32] J.E. Hutchinson. The geometry of fractal sets. Indiana University Mathematics Journal, 30:713–747, 1981.

[33] O. Lanford III. A computer-assisted proof of the feigenbaum conjectures. Bull. Amer. Math. Soc., 6:427–434, 1982.

[34] O. Lanford III. Introduction to hyperbolic sets. In Regular and Chaotic Motion in Dynamical systems, volume 118 of Nato ASI Ser. B, pages 74–102, 1985.

[35] O. Lanford III. Lectures on dynamical systems. Notes from lectures held at ETH, 1990-1992, 1992.

[36] A. Jakobsen and O. Knill. Iteration of the coupled map lattice construction. Phys. Lett. A, 205:179–183, 1995.

[37] K. Kaneko. Collapse of tori and genesis of chaos in dissipative systems. Ph.D thesis, (enlarged version is published by World Scientific, 1986), 1983.

[38] K. Kaneko. Chaos, focus issue on coupled map lattices. Chaos, 2, 1992.

[39] K. Kaneko. Theory and applications of coupled map lattices. Chichester : New York : John Wiley and Sons, 1993.

[40] A. Katok and B. Hasselblatt. Introduction to the modern theory of dynamical systems, volume 54 of Encyclopedia of Mathematics and its applications. Cambridge University Press, 1995.

[41] F.R. Marotto. Chaotic behavior in the Hénon mapping. Commun. Math. Phys., 68:187– 194, 1979.

[42] J. Mather. Glancing billiards. Ergod. Th. Dyn. Sys., 2:397–403, 1982.

[43] R. McGehee and K.R.Meyer (Editors). Twist Mappings and their applications. IMA Volumes in Mathematics, 1992. Vol. 44.

[44] M. McGuire. An eye for fractals. Addison-Wesley, Redwood City, 1991.

[45] C.T. McMullen. Complex Dynamics and Renormalization, volume 135 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1994.

[46] J. Milnor. Non-expansive Hénon maps. Adv. in Math., 69:109–114, 1988.

[47] J. Milnor. Dynamics in one complex variable. Introductory Lecures, SUNY, 1991.

[48] J. Moser. Stable and random Motion in dynamical systems. Princeton University Press, Princeton, 1973.

[49] S. Nakane. Connectedness of the tricorn. Ergod. Th. Dyn. Sys., 13:349–456, 1993.

[50] H-O. Peitgen, H.Jürgens, and D.Saupe. Chaos and Fractals. Springer-Verlag, New York, 1992.

[51] H-O. Peitgen and P.H.Richter. The beauty of fractals. Springer-Verlag, New York, 1986.

[52] C.A. Rogers. Hausdorff measures. Cambridge university press, 1970.

[53] D. Ruelle. Chaotic evolution and strange attractors. Cambridge University Press, Cambridge, 1987.

[54] D. Ruelle. Chance and chaos. Princeton University Press, Princeton, 1991.

[55] H.G. Schuster. Deterministic Chaos. VCH Verlagsgesellschaft, 1989.

[56] C. Sparrow. The Lorenz Equations, bifurcations, chaos and strange attractors, volume 41 of Applied Mathematical Sciences. Springer-Verlag, 1982.

[57] I. Stewart. Does God play dice. Blackwell, Cambridge, 1989.

[58] D.V. Treshchev V.V. Kozlov. Billiards, volume 89 of Translations of mathematical monographs. AMS, 1991.

[59] M. Wojtkowski. Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys., 105:391–414, 1986.

[60] S. Wolfram. Theory and Applications of Cellular Automata. World Scientific, 1986.

[61] J.C. Yoccoz. Introduction to hyperbolic dynamics. Lecture notes from NATO Advanced Study Institute, June 20- July 2, 1993.

[62] S.L. Ziglin. Nonintegrability of a problem of the motion of four point vortices. Sov. Math. Dokl., 21:296–299, 1980.

Index

ABC flow, 164 Abel-Jacobi map, 184 ABS flow, 28 Algebraic Structure, 11 Alphabet

of shift, 33 Analytic map, 88 Analytic standard map, 114 Annulus, 69 Anosov property, 19, 115 Antoine Necklace, 57 Asymptotically stable, 154 Attracting set, 158 Attractor, 160 Aubry-Mather sets, 113 Automaton

Shift register, 130 Axiom A, 19

Banach space, 35 Band spectrum

periodic Jacobi matrices, 182 Barnsley fern, 56 Barnsley’s fern, 25 Basin of attraction, 41, 99 Belousov-Shabotinsky system, 163 Bernoulli property, 19 Bernoulli shift, 21, 33 Bifurcation Diagram, 22 Billiard map, 23, 69 Billiard map

Ergodic, 75 Billiard table, 68 Birkhoff conjecture, 81 Birkhoff periodic orbits, 113 Bobenko-Pinkall map, 105 Bounded Distortion, 51 Box counting dimension, 57, 65 Brooks, 6 Brooks-Matelski set, 6 Busy beaver, 25 Butterfly effect, 5

Cantor set, 54

Caranicolas-Vozikas system, 28 Cauchy sequence, 34 Caustic, 75 Cellular automata, 23 Cellular automaton, 123 Cellular automaton Bobenko, 129 Bobenko-Pinkall, 130 Conway, 130 Elementary, 128 Hexagonal lattice model, 131 HLG, 131 HPP, 131 Life, 130 reversible, 129 Wolfram, 128

Chaologist, 4

Chaos

What is chaos, 4 Chaotic

in the sense of Devaney, 33 in the sense of Lasota, 18

Chaotic in the sense of Devaney, 18

in the sense of Li-Yorke, 18 Chiricov map, 31, 103 Circle map, 22 Colin de Verdiere, 31 Collar, 52 Collatz conjecture, 194 Collatz map, 24, 195 Collatz problem, 195 Complete, 34 Completely invariant set, 98 Complex Analytic Structure, 12 Connected topologica space, 89 Contopoulos-Barbanis system, 28 Contraction, 34 Contractivity Factor, 34 Conway map, 25 Coupled map lattice, 142, 147 Cover, 48 Curvature, 68 Curve

Closed, 68 Convex, 68 Definition, 68 Smooth, 68

Data encryption standard, 199 Degree of a rational map, 96 Dependent Percolation, 140 Derivative

Fréchet, 35 DES, 199 Diameter of set, 48 Differentials of the first kind, 184 Dimension

Box counting, 57, 65

Entropy, 65 Fractal, 65 Hausdorff, 49, 64 Inductive, 64 Similarity, 65 Topological, 49, 64

Discrete Logarithm problem, 200 Discrete Ricatti equation, 182 Distal, 17 Domain, 89 Domain of instability, 75 Douady, 6 Duffing equation, 27 Dynamical system, 9 Dynamical system , general definition10

Continuous time, 10 Discrete time, 10

Elliptic periodic orbit, 72 Entropy dimension, 65 Ergodic, 18 Ergodic billiard map, 75 Euler equations, 190 Exceptional set, 99 Expandivity, 19 Expansivity, 17 Exterior Billiard map, 80

Fatou set, 84 Feigenbaum, 7 Feigenbaum attractor, 7 Filled-in Julia set, 87 Fixed point

Attractive, 91 Repelling, 91 Super attractive, 91

Formula

Green MacKay,Meiss, 74 Fréchet

Derivative, 35 Fractal

Definition, 49 Fractal dimension, 65 Fractals

What are fractals, 6 Francescini and Russo, 42

Game of Life, 23

Generating function , Billiard map70 Generalized standar map, 106 Héneon map, 71 Geometric Structure, 11 Green function, 92 Group Automorphism, 21 Group translation, 21

Hénon attractor, 41 Hénon map, 7, 31 Hénon-Heils system, 27 Hausdorff dimension, 49, 64 Hausdorff distance, 52 Hausdorff measure, 48 Heat flow, 29 Hessian, 108 Homomorphism of Dynamical systems, 14 Hopf flow, 28 Hubbard, 6 Hutchinson map, 25 HUtchinson operator, 53 Hutchinson operator

on measures, 62 Hyperbolic

For smooth dynamical systems, 19 Hyperbolic periodic orbit, 72 Hyperbolic set, 115 Hyperbolicity

Nonuniform, 19

Topological dynamical system, 18 Hyperelliptic curve, 184

Immediate basin of attraction, 99 Implicit Function theorem, 37 Inductive dimension, 64 Integrated Euler equations, 190 Interval Map, 22 Invariant subset, 32 Isomorphism of Dynamical systems, 14 Iterated function system, 53

Jacobi matrix, 178 Jacobian, 32 Japanese attractor, 27, 164 Julia set, 6, 56, 84

Koch curve, 54 Koch iland, 54 Koch snowflake, 54 Kolmogorov property, 19

Lax pair, 178

Lebesgue property, 18

Lemma Böttcher-Fatou, 91 Wojtkowski, 76 Lipeomorphism, 51 Lipshitz continuous, 51 Locally connected, 90 Logistic Map, 85 Lorenz equations, 150 Lorenz system, 8, 26, 149 Lozi map, 31 Lyapunov function, 153 Lyapunov stable, 154

Mandelbar set, 94 Mandelbrot, 6 Mandelbrot set, 6, 84, 87 Map

Billiard, 69

Bobenko-Pinkall, 105 Chiricov, 103 Exterior biliard, 80 Logistic, 85 Moser, 80 Quadratic, 84 Standard, 103 Wojtkowsky, 105

Markov Chain, 21 Martin map, 31 Matelski, 6 Mathematical Pendulum, 27 Mathematical Structure, 9 Mathematical structures

Basic, 10 Maximum principle, 90 Measure theoretical Structure, 11 Menger sponge, 55 Meromorphic map, 88 Metric isomorphism, 51 Minimal, 17 Mixing, 18 Moduli space, 9 Monotone Twist map, 23 Moran’s open set condition, 59 Moser map, 80 Multiple Birkhoff recurrence theorem, 140

Navier-Stokes flow, 29 Neural network, 145 Noncommutative Mathematics, 12 Nonlinear Pendulum, 191 Nonuniform Hyperbolicity, 19 Norm, 35 Normal number, 197 Normed Vector Space, 35

Omega limit set, 153 Orbit, 32 Orszag-McLaughlin flow, 29

Parabolic periodic orbit, 72 Pedal map, 24 Percolation

dependent, 140 Peridic Jacobi matrix, 73 Period

orbit, 32 Periodic orbit, 39

billard, 71 Periodic Orbit

Attracting, 39 Repelling, 39

Periodic Orbits, 15 Periodic orbits

Birkhoff, 113 Permutation, 20 Phase space reconstruction, 161 Poincaré map, 30 Poisson bracket, 173 Potential

For twist map, 31 Prandl number, 150 proper, 94 Pseudo random generator, 198

Quadratic Familiy, 23 Quadratic map, 84

Róssler system, 163 Radius of Cellular automaton, 124 Random walk, 24 Rannou, 20 Rational map, 23 Recurrence theorem

Birkhoff, 140 Region, 89 Renormalisation maps, 26 Retarded differential equations, 30 Ricatti equation, 182 Roessler attractor, 26 Roessler system, 26 Ruelle, 7 Runge-Kutta integration, 149

Sahos-Bountis system, 28 Schrödinger flow, 29 Semi orbit, 153 Semigroup, 10 Sensitive dependence on initial conditions,

17 Shift, 33, 123 Sierpinski carpet, 55 Sierpinski gasket, 55 Similarity dimension, 59, 65 Similarity transformation, 58 Simply connected, 89 Smith, 6 Solenoid, 56 Space-time chaos, 7 Stórmer system, 27 Stable, 154 Stable manifold, 39 Standard map, 103 Stoboscopic map, 30 Strange attracting set, 160 Strange attractor, 7, 160 Strictly ergodic, 17 Sturmean systems, 134 Substitution, 134 Subsystem of a Dynamical system, 15 Suris map, 31 Suspension flow, 30

Takens, 7 Theorem

Banach fixed point theorem, 34 Birkhoff, 71 Cauchy-Picard Existence theorem, 151 Devaney-Nitecki, 38 Douady-Hubbard, 90 Elton, 64 Existence of stable and unstable man-

ifolds, 40 Fürstenberg-Kesten, 76 Grobmann-Hartman, 155 Gruenwald, 202 Hedlund-Curtis-Lyndon, 124 Hurd-Kari-Culik, 127 Montel, 98 Poincaré recurrence, 70 Poincaré-Bendixen, 153 Shishikura, 100 Sigmund, 134 van der Waerden, 140 Wojtkowski, 77

Theorem of Matthews-Watts, 196 Theroem

Multiple Birkhoff recurrence, 201 Time one map, 30, 159 Toda system, 28, 177 Toepitz systems, 135 Topological dimension, 64 Topological dynamical system, 32 Topological entropy, 115 Topological Structure, 11 Topologically Mixing, 17 Totally ergodic, 18 Transfer matrix, 73 Transitiv, 17 Trapping region, 41, 152 Tricorn, 94 Turbulence

What is turbulence, 7 Turing machine, 136 Turing machines, 25 Twist map

Chiricov, 31

Colin de Verdiere, 31 Hénon, 31 Lozi, 31 Martin, 31 Plane, 31 Suris, 31 Wojtkowsky, 31

Ueda attractor, 27, 164 Ueda system, 27 Ulam, 5 Ulam conjecture, 194 Uniquely ergodic, 17 Unstable Manifold, 39 Upper and lower Density, 50 Upper capacity, 57

Van Allen radiation belts, 28 Van der Pool oscillator, 27 Van der Waerden’s theorem, 140 Von Neumann, 5 Vortex, 171 Vortex flow, 29

Weakly mixing, 18 Weighted iterated function system, 62 Wojtkowsky map, 31, 105