Inventiones mathematicae

, Volume 113, Issue 1, pp 617–631

# Quantitative recurrence results

• Michael D. Boshernitzan
Article

## Summary

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-measureHα(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.

## Keywords

Probability Measure Additional Assumption Countable Base Recurrence Result Quantitative Recurrence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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