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

Published online by Cambridge University Press:  10 August 2015

FRANÇOISE PÈNE  and
BENOÎT SAUSSOL
FRANÇOISE PÈNE
Affiliation:
Université de Brest, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS UMR 6205, Brest, France email [email protected], [email protected]
BENOÎT SAUSSOL
Affiliation:
Université de Brest, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS UMR 6205, Brest, France email [email protected], [email protected]
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2602 - 2626
Copyright
© Cambridge University Press, 2015 

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