Quantitative recurrence results

Summary LetX be a probability measure spaceX=(X, , ) endowed with a compatible metricd so that (X,d) has a countable base. It is well-known that ifTXX is measure-preserving, then -almost all pointsxX are recurrent, i.e., . We show that, under the additional assumption that the Hausdorff -measureH (X) ofX is -finite for some >0, this result can be strengthened: , for -almost all pointsxX. A number of applications are considered.Oblatum 24-II-1992 & 8-II-1993Supported in part by NSF-DMS-9003450