By Steven Kalikow

An advent to ergodic idea for graduate scholars, and an invaluable reference for the pro mathematician.

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For a development of the theory of such systems, including a relevant extension of the Birkhoff ergodic theorem, see Gray (1988, Chapters 6–8). 208. Exercise. ) In the ergodic case of N f (T i ω) = f dμ. Hint: first show that Theorem 202, lim N →∞ N1 n=1 35 The argument runs as follows. For ω ∈ S , one has M−1 i=0 1 f (T i ω) = M x∈Iω 1 ≥ M ≥ |E1 | ω 1 f 1 (x) + M f 2 (x) x∈Iω 1 | f 1 (x)| − M | f 2 (x)| x∈Iω \E ω x∈Iω B N + b−a M − j (ω) f 1 (x) − M M 12B f 1 (x) − x∈E ω x∈E ω b−a b−a b−a b+a ≥ b − a − b−a 4 − 24 − 12 − 8 = 2 .

5. Countable generator theorem 37 185. Exercise. Show that {S, T S, T 2 S, . . , T N −1 S} is a Rohlin tower of height N , and that, if δ is small enough and M large enough, one may, by shaving off a small part of S and throwing it into the error set, achieve independence from P while keeping the error set under in measure. 186. Definition. The superimposition of a sequence of partitions (Pi ) is the partition of into the equivalence classes of ∼, where x ∼ y if for every i ∈ N, x and y are in the same cell of Pi .

Sketch of proof. Now for the details. Fix > 0 and a finite word w. For m ∈ N let Bm = y ∈ : there exists m ≥ m such that r f w; wm (y) −r f (w; y) > . Notice that B m+1 ⊂ B m . 230. Exercise. Use the Birkhoff ergodic theorem applied to the function 1ϕ(w) to show that limm→∞ μ(B m ) = 0. e. by Birkhoff. 8. Ergodic decomposition 231. Exercise. Use the dominated convergence theorem to show μ(B m ). 47 lm dμ = • Now put E m = {x : l(x) ≥ }. 232. Exercise. Show that E m+1 ⊂ E m . Show also that if δ > 0 and μ(B m ) < • δ then μ(E m ) < δ.