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Description

We are interested in the counting process of visits to a small set, and more precisely in its behaviour as the measure of the set goes to 0. We prove the convergence in distribution of this process to a Poisson process under general assumptions. We apply our general results in different expanding/hyperbolic contexts. We study in particular the case of chaotic billiard ows: Sinai billiard, the Bunimovich billiard in a stadium, billiards with corners and without cusps. We obtain a general result for visits to balls around a generic point when the system is modeled by a Markov Young tower, and also for visits to balls around an hyperbolic periodic point. This is a joint work with Benot Saussol, inspired by a question by Domokos Szasz and Peter Imre Toth.