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

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 particu...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2013-02, Vol.33 (1), p.49-80
Main Authors: CHAZOTTES, J.-R., COLLET, P.
Format: Article
Language:English
Subjects:
Dynamical systems
Poisson distribution
Theorems
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385711000897