By Steven Kalikow

This casual creation specializes in the department of ergodic thought often called isomorphism thought. workouts, open difficulties, and valuable tricks actively interact the reader and inspire them to take part in constructing proofs independently. perfect for graduate classes, this booklet can also be a important reference for the pro mathematician.

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Additional info for An outline of ergodic theory

Example text

223. Comment. In order to create a stationary measure via the monkey method from a generic point, it isn’t necessary to pass to subsequences. Note that if μ is the resulting measure, r f (w, x) = μ ϕ({w}) for every finite word w. We express this by saying that x is generic for μ. It is important to note that the notion of a generic point is not more general than that of one generic for some μ; every generic sequence is generic for its own monkey measure. 8. 8. Ergodic decomposition In this subchapter, we show how to decompose an arbitrary measurepreserving system as an integral of ergodic ones.

Let ( , A, μ, T ) be a non-periodic measure-preserving system, let N ∈ N and let > 0. For any finite measurable partition P, there exists some S ∈ A that 32 That is, E = {x ∈ \ ∞ i j i=−∞ T C : |x − T x| > δ, 1 ≤ j ≤ M} and one has μ(E) > 0. 9780521194402c02 CUP/KKW 36 October 10, 2009 21:55 Page-36 Measure-preserving systems is independent of P such that S, T S, T 2 S, . . , T N −1 S are pairwise disjoint N −1 i T S) = . and μ(X \ i=0 Sketch of proof. 33 Let δ be extremely small and let M be extremely large.

X −3 , X −2 , X −1 , X 0 , X 1 , X 2 , X 3 , . . 25 ∞ 152. Discussion. Given a stationary process (X i )i=−∞ , where the X i take values in a countable alphabet = {λ1 , λ2 , . }, one can form the associated measure-preserving shift system ( , A, μ, T ) per Exercise 99, then let P be the partition consisting of the pieces {x : x0 = λi }, i = 1, 2, . , and construct the (P, T ) process, which will be isomorphic to the original process and, for all practical purposes, indistinguishable from it.