Hyperbolic manifolds, discrete groups and ergodic theory

Math 277 – Fall 1996 – Berkeley C. McMullen

Homework 1 Due Tuesday, 1 October 96

Attempt as many problems as you wish. Cite references or collaborators if used. Problems marked with (*) may be especially challenging.

1. Give an explicit Borel automorphism of [0, 1] sending the standard probability measure on the Cantor set C to Lebesgue measure on [0, 1]. (In the standard measure on C, the digits in the base 3 expansion of x ∈ C are independent random variables taking the values 0 and 2 each with equal probability 1/2.)

2. Let T : S^{3} → S^{3} be an isometry. Can T have a dense orbit on S^{3}? What sets can arise as the closure of an orbit {T^{n}(x) : x ∈ S^{3}}?

3. Let T : X → X be ergodic, with invariant probability measure m. Let A ⊂ X have positive measure, and let k : A → ℕ be defined by k(x) = inf{i > 0 : T^{i}(x) ∈ A}. (Note that k(x) is finite a.e.)

Show that the average of k(x) over A is 1/m(A). In other words, the average return time to A is inversely proportional to the measure of A.

4. Let T : S^{1} → S^{1} be the irrational rotation T (z) = exp(2πiγ)z. Let I ⊂ S^{1} be a fixed interval, 0 < m(I) < 1, and let

a_{n}(z) = \frac{1}{n + 1} ∑_{0}^{n} χ_{I}(T^{i}z).

Let E_{n} = sup_{z} |a_{n}(z) − m(I)| denote the maximum error obtained if you estimate the measure of I by the frequency of visits of the orbit of z.

(a) Show a_{n}(z) converges uniformly to m(I); that is, E_{n} → 0.

(b) Show that for any sequence ϵ_{n} → 0, there is a γ such that E_{n} > ϵ_{n} infinitely often.

(c*) Estimate the size of E_{n} when γ = (\sqrt{5} − 1)/2 is the golden ratio. (Hint: |γ − p_{i}/q_{i}| < C/q_{i}^{2} where ⟨q_{i}⟩ = ⟨1, 2, 3, 5, 8, 13, …⟩ are the Fibonacci numbers, and p_{i} = q_{i−1}.)

5. Let T : S^{1} → S^{1} be an irrational rotation. Show there is an f ∈ C(S^{1}) with ∫ f = 0 such that f is not a coboundary; i.e., such that f ≠ g−g∘T for any g ∈ C(S^{1}).

6. Show that a toral automorphism specified by T ∈ SL_{n}ℤ is ergodic if and only if T has no eigenvalue which is a root of unity.

7. Prove any toral automorphism T has a dense set of periodic points.

8. Prove that any “trigonometric polynomial” f(z) = ∑_{−N}^{N} a_{n}z^{n}, such that f(z) ≥ 0 when |z| = 1, satisfies form f(z) = |g(z)|^{2} on S^{1} for another trigonometric polynomial g(z). (Hint: write f(z) as a quotient of polynomials P (z)/Q(z).)