Hausdorff dimension of measures via Poincare recurrence

We study the quantitative behavior of Poincare recurrence. In particular, for an equilibrium measure on a locally maximal hyperbolic set of a C1+alpha diffeomorphism f, we show that the recurrence rate to each point coincides almost everywhere with the Hausdorff dimension d of the measure, that is, inf{k > 0 : f(k)x epsilon B(x, r)} similar to r(-d). This result is a non-trivial generalization of work of Boshernitzan concerning the quantitative behavior of recurrence, and is a dimensional version of work of Ornstein and Weiss for the entropy. We stress that our approach uses different techniques. Furthermore, our results motivate the introduction of a new method to compute the Hausdorff dimension of measures.