An outline of ergodic theory by Steven Kalikow

By Steven Kalikow

This casual creation makes a speciality of the department of ergodic thought often called isomorphism concept. routines, open difficulties, and beneficial tricks actively have interaction the reader and inspire them to take part in constructing proofs independently. excellent for graduate classes, this publication is additionally a beneficial reference for the pro mathematician.

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Assume the heights of these towers are N1 , N2 , . . Let Pi be a partition of the whole space, consisting of the complement of Si together with the following partition of Si : two points x and y are in the same atom if and only if for all k ∈ {1, 2, . . 6. Birkhoff ergodic theorem and the strong law Page-39 39 Ni digits in their binary expansions. The superimposition of the Pi supplies the desired partition; error sets can be obnoxious but Borel–Cantelli can dispose of all but finitely many of them.

Then let Yi = f ◦ X i . ∞ is an independent stationary process on a countable 211. Exercise. (Yi )i=−∞ alphabet. • Let ( , A, μ, T ) be the measure-preserving system associated with (Yi ). By Theorem 149, ( , A, μ, T ) is ergodic. Define a measurable function ∞ = f (x0 ). g : → R by g (xi )i=−∞ 212. Exercise. Show that g dμ = ∞ i i=−∞ n P(Y0 = ni ) = E(Y0 ). e. ω ∈ . 1 N N n=1 g(T i ω) • = ∞ 213. Exercise. Define π : Z → by π(z) = Yi (z) i=−∞ . e. for every i ∈ Z. e. and use this to complete Conclude that lim N →∞ N1 n=1 the proof.

Exercise. There exists δ > 0 such that, with positive probability, x is not in the union of the translates of C and T x, T 2 x, . . , T M x are all more than δ away from x. 32 Hint: let Fn = {x : |x − T j x| > n1 , 1 ≤ j ≤ • M} and show that μ(Fn ) → 1. Choose an interval I of diameter < δ such that μ(I ∩ E) > 0. Let C = C ∪ (I ∩ E). 177. Exercise. Show that C ∈ S, C C and C ∼ C, a contradiction. 178. Definition. Let ( , A, μ, T ) be an invertible measure-preserving system and suppose S ∈ A with S, T S, T 2 S, .

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