Quantitative recurrence results

Abstract

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., {Mathematical expression}. We show that, under the additional assumption that the Hausdorff α-measure Hα(X) of X is σ-finite for some α>0, this result can be strengthened: {Mathematical expression}, for μ-almost all points x∈X. A number of applications are considered. © 1993 Springer-Verlag.