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We study the number of visits to balls B_r(x), up to time t/mu(B_r(x)), for aclass of non-uniformly hyperbolic dynamical systems, where mu is the SRBmeasure. Outside a set of `bad' centers x, we prove that this number isapproximately Poissonnian with a controlled error term. In particular, whenr-->0, we get convergence to the Poisson law for a set of centers of mu-measureone. Our theorem applies for instance to the H\'enon attractor and, moregenerally, to systems modelled by a Young tower whose return-time function hasa exponential tail and with one-dimensional unstable manifolds. Along the way,we prove an abstract Poisson approximation result of independent interest.

Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems