Three Results in Recurrence

Proofs of some recent results of Kriz and Forrest are presented which distinguish between four diierent kinds of \sets of recurrence" arising naturally in connection with topological and measure-preserving dynamical systems. 0. Suppose that (X; ; ; T) is an invertible measure preserving system, that is, (X; ;) is a measure space with (X) = 1 and T : X ! X is a bijection mod 0, with (A) = (TA) = (T 1 A) for all measurable sets A. (By bijection mod 0 we mean that there exists X 0 2 , (X 0) = 1, such that the restriction of T to X 0 is a bijection.) Then Theorem 0.1. (Poincar e) If A 2 with (A) > 0, then for some natural number n, (A \ T n A) > 0. we have (A \ T k A) > 0. Proof. The sets T n A, n 2 E, being of equal positive measure, cannot be pairwise disjoint in the nite measure space X, hence for some n; m 2 E, n > m, (T n A \ T m A) > 0. Since T is measure preserving we have (A \ T nm A) > 0. Theorem 0.2 motivates the following deenition.