We study the quantitative behavior of Poincare recurrence. In particular, f
or 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-t
rivial generalization of work of Boshernitzan concerning the quantitative b
ehavior of recurrence, and is a dimensional version of work of Ornstein and
Weiss for the entropy. We stress that our approach uses different techniqu
es. Furthermore, our results motivate the introduction of a new method to c
ompute the Hausdorff dimension of measures.