## Summary

Let*X* be a probability measure space*X*=(*X*, Φ, μ) endowed with a compatible metric*d* so that (*X*,*d*) has a countable base. It is well-known that if*T*∶*X*→*X* is measure-preserving, then μ-almost all points*x*∈*X* 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-measure**H**_{α}(*X*) of*X* 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 points*x*∈*X*. A number of applications are considered.

## Keywords

Probability Measure Additional Assumption Countable Base Recurrence Result Quantitative Recurrence## Preview

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