Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems

TL;DR

This paper proves that in certain non-uniformly hyperbolic dynamical systems, the distribution of return times to metric balls converges to a Poisson distribution with explicit error bounds, applicable to SRB measures.

Contribution

It establishes Poisson limit laws for return times in systems with polynomial decay of correlations, providing explicit error terms and conditions for uniform convergence.

Findings

Return times to metric balls are Poisson distributed in these systems.

Error terms decay as powers of logarithm of the radius.

Results apply to SRB measures on attractors.

Abstract

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.