
ORBITS OF AUTOMORPHISMS OF K3 SURFACES

	Version 3.2	3/2006

QUICKSTART.
Type "cd src; make" to install,
then "cd ../example; ./Traj.run; ./Stable.run" to test.

THEORY.  A K3 surface S in R^3 is specified by the equation

(1+x^2)(1+y^2)(1+z^2) + Axyz = B .

Each line parallel to the x, y or z axis meets S in 2
points, so we obtain 3 involutions by "sheet interchange".
There is also a useful symmetry G(x,y,z) = (x,-y,-z).
The dynamical system we investigate is f : S -> S given by 

	f(p) = G I_x I_y I_z (p).

This map preserves a natural volume element on S.

PROGRAMS.  This directory contains 2 programs:
	traj	- plots orbits of f(p);
	stable  - plots its stable manifold.

TRAJ OUTPUT.  The output of "traj" is a .gif file.
The main program, "trajps", writes postscript, which
is piped to obtain a .gif image.
The postscript output can also be used directly.

TRAJ INPUT.  All input to traj is via command-line arguments.
The arguments are processed in order, producing
an image that may include many different features.

-c [x y] -r [r]
	Set window for image by giving center and "radius".
	Default is center (0,0), radius 1.2.

-d [x y]
	Dimension of raster image to be output.
	The square specified by the window will always
	be included.

-i [n]
	Iterations - number of points in orbits.
	Default 10,000.

-o [x y]
	Draw the orbit with given real coordinates.
	Traj solves for z such that p=(x,y,z) lies on
	the surface S.  

-p [A B]
	Change the parameters for the surface S.

-R [n]
	Draw n random orbits.

-t
	Toggle top sheet filter.  By default, the
	surface S is "opaque", and only points 
	on the top sheet are drawn.   This filter
	make S "transparent", so orbits on both
	sheet of S appear.

STABLE OUTPUT.  The output of "stable" is a .ps file.
The map f has a hyperbolic fixed-point at (0,0,1), and
stable draws its stable manifold.

EXAMPLES.  See the example/ directory for usage.

STABLE INPUT.  All input to stable is via command-line arguments.

-b	Draw only bottom of surface.  By default the surface is
	transparent.

-c [x y] -r [r]
	Set window for image by giving center and "radius".
	Default is center (0,0), radius 1.2.

-e [eps]
	Set the accurancy of the stable manifold plot. 
	The manifold is drawn as a series of line segments each
	of length about eps.

-l [L]
	Continue extending the stable manifold until the
	path joining p to f(p) has length > L.

-p [A B]
	Change the parameters for the surface S.

-q	Quiet.

-s	Spin surface (draw image under (xyz) -> (yzx))

-t	Draw only top of surface

EXAMPLES.  See the example/ directory for usage.


Copyright (c) March 2006

Curtis T McMullen

ctm@math.harvard.edu

