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We consider some nonuniformly hyperbolic invertible dynamical systems whichare modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for theinducing time and a polynomial control of hyperbolicity, as introduced byAlves, Pinheiro and Azevedo. These systems admit a physical measure withpolynomial rate of mixing. In this paper we prove that the distribution of thenumber of visits to a ball B(x, r) converges to a Poisson distribution as theradius r $\to$ 0 and after suitable normalization.

Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing