We show that for systems that allow a Young tower construction with
polynomially decaying correlations the return times to metric balls are in the
limit Poisson distributed. We also provide error terms which are powers of
logarithm of the radius. In order to get those uniform rates of convergence the
balls centres have to avoid a set whose size is estimated to be of similar
order. This result can be applied to non-uniformly hyperbolic maps and to any
invariant measure that satisfies a weak regularity condition. In particular it
shows that the return times to balls is Poissonian for SRB measures on
attractors.Comment: 28 page