Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing

Abstract

We consider some non-uniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs–Markov–Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball$B(x,r)$converges to a Poisson distribution as the radius$r\rightarrow 0$and after suitable normalization.