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Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

Published online by Cambridge University Press:  07 February 2012

J.-R. CHAZOTTES  and
P. COLLET
J.-R. CHAZOTTES
Affiliation:
Centre de Physique Théorique, CNRS–École Polytechnique, 91128 Palaiseau Cedex, France (email: [email protected], [email protected])
P. COLLET
Affiliation:
Centre de Physique Théorique, CNRS–École Polytechnique, 91128 Palaiseau Cedex, France (email: [email protected], [email protected])
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Abstract

We study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of ‘bad’ centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r→0, we get convergence to the Poisson law for a set of centers of μ-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has an exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 49 - 80
Copyright
Copyright © Cambridge University Press 2013

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