By Karl E. Petersen

ISBN-10: 0521389976

ISBN-13: 9780521389976

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**Additional resources for Ergodic Theory**

**Example text**

3. Recurrence 37 ing to the Recurrence Theorem, IwEE: there is k> 0 such that crk coeE} has measure equal to p(E). That is. with probability 1 the system will recur again and again to the state in which there are no balls in the first urn. 6), it is 1/p(E) = 2 1 " drawings. For real systems with millions of particles, the expected return times will be even more absurd. For recurrence of sets a sharper, quantitative statement can be made. A set E c Z is called relatively dense if it has bounded gaps.

E. e. x Proof: Suppose (X, gi1, p, T) is ergodic. e. e. e. e. 1 implies that (X, a, p, T) is ergodic. For a measurable set E and a point xe X. con y xE (Tkx) =1E(x). 4. e. e. , so ,u(E) must be 0 or 1. Thus T is ergodic if and only if the mean sojourn time in a measurable set equals the measure of the set for almost all points of X. The theory of Krylov and Bogoliouboff (1937) (see also Oxtoby 1952) reverses this situation by starting with a measurable transformation and trying to define an ergodic invariant measure by 1'' p(E)= lim 11 -* C0 E XE (T kX) n k =0 when the limit exists.

It may be considered to be the most basic result in ergodic theory. 1 Let BEM. A point xeB is said to be recurrent with respect to B if there is a k 1 for which rx GB. 2 Poincaré Recurrence Theorem (1899) For each BEM, almost every point of B is recurrent with respect to B. Proof: Let F be the set of all those points of B which are not recurrent with respect to B; then F = B— k=1 T -k B=BnT -1 (X — B)nT -2 (X — B)n... 1cOF for each n 1. Thus F nT'F = 25 for n 1. and hence T'FnT -(n +k) F Ø for each n 1 and each k O.

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