Inventiones mathematicae

, Volume 113, Issue 1, pp 617-631

First online:

Quantitative recurrence results

  • Michael D. BoshernitzanAffiliated withDepartment of Mathematics, Rice University

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LetX be a probability measure spaceX=(X, Φ, μ) endowed with a compatible metricd so that (X,d) has a countable base. It is well-known that ifTXX is measure-preserving, then μ-almost all pointsxX are recurrent, i.e.,\(\lim \begin{array}{*{20}c} {\inf } \\ {n \geqq 1} \\ \end{array} d(x, T^n (x)) = 0\). We show that, under the additional assumption that the Hausdorff α-measureH α(X) ofX is σ-finite for some α>0, this result can be strengthened:\(\lim \begin{array}{*{20}c} {\inf } \\ {n \geqq 1} \\ \end{array} \left\{ {n^{1/\alpha } . d(x, T^n (x))} \right\}< \infty \), for μ-almost all pointsxX. A number of applications are considered.