An outline of ergodic theory - download pdf or read online

By Steven Kalikow

ISBN-10: 0511676972

ISBN-13: 9780511676970

ISBN-10: 0511679483

ISBN-13: 9780511679483

ISBN-10: 0511681461

ISBN-13: 9780511681462

ISBN-10: 0511801602

ISBN-13: 9780511801600

ISBN-10: 0521194407

ISBN-13: 9780521194402

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

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G. 14]), systems with continuous time (see Krengel 1985, p. 10, for discussion), as well as for some systems without an invariant measure. Indeed, a (possibly non-measure preserving) system ( , A, μ, T ) is said to be asymptotn μ(T −i A) exists for every ically mean stationary, or AMS, if limn n1 i=1 measurable set A. 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.

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.

X, A, μ, T ) is an ergodic system if and only if for every finite word w one has n 1ϕ(w) (T i x) = μ ϕ(w) in measure. e. to f and in measure to g then f = g. 226. Theorem. ) and x ∈ is generic for μ and has the following property P(x): P(x): For any k ∈ N and any word w ∈ Wk there exists c(w, x) such that for every > 0 there is an M0 such that if m > M0 there exists some N0 such that for all n > N0 , if we let B = B(m, w, x) = {u ∈ Wm : |r f (w; u) − c(w, x)| ≥ } then r f B, wn (x) < . Then ( , A, μ, T ) is ergodic.

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An outline of ergodic theory by Steven Kalikow

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