Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T16:53:20.476Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abadi, M. and Vergne, N.. Sharp error for point-wise Poisson approximations in mixing processes. Nonlinearity 21 (2008), 2871–2885.CrossRefGoogle Scholar
[2]Arratia, R., Goldstein, L. and Gordon, L.. Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Probab. 17(1) (1989), 9–25.Google Scholar
[3]Bruin, H., Saussol, B., Troubetzkoy, S. and Vaienti, S.. Return time statistics via inducing. Ergod. Th. & Dynam. Sys. 23(4) (2003), 991–1013.CrossRefGoogle Scholar
[4]Bruin, H. and Vaienti, S.. Return time statistics for unimodal maps. Fund. Math. 176(1) (2003), 77–94.Google Scholar
[5]Chazottes, J.-R., Collet, P. and Schmitt, B.. Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems. Nonlinearity 18 (2005), 2341–2364.CrossRefGoogle Scholar
[6]Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21(2) (2001), 401–420.CrossRefGoogle Scholar
[7]Collet, P. and Galves, A.. Statistics of close visits to the indifferent fixed point of an interval map. J. Stat. Phys. 72(3–4) (1993), 459–478.Google Scholar
[8]Collet, P. and Galves, A.. Asymptotic distribution of entrance times for expanding maps of the interval. Dynamical Systems and Applications (World Scientific Series in Applied Analysis, 4). World Scientific, River Edge, NJ, 1995, pp. 139–152.CrossRefGoogle Scholar
[9]Denker, M., Gordin, M. and Sharova, A.. A Poisson limit theorem for toral automorphisms. Illinois J. Math. 48(1) (2004), 1–20.CrossRefGoogle Scholar
[10]Dolgopyat, D.. Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356(4) (2004), 1637–1689.CrossRefGoogle Scholar
[11]Federer, H.. Geometric Measure Theory (Die Grundlehren der mathematischen Wissenschaften, 153). Springer, New York, 1969.Google Scholar
[12]Freitas, A. C., Freitas, J. M. and Todd, M.. Hitting time statistics and extreme value theory. Probab. Theory Related Fields 147(3–4) (2010), 675–710.Google Scholar
[13]Gupta, C., Holland, M. and Nicol, M.. Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2009.Google Scholar
[14]Haydn, N. and Vaienti, S.. The limiting distribution and error terms for return times of dynamical systems. Discrete Contin. Dyn. Syst. 10(3) (2004), 589–616.CrossRefGoogle Scholar
[15]Hirata, M.. Poisson law for Axiom A diffeomorphisms. Ergod. Th. & Dynam. Sys. 13 (1993), 533–556.Google Scholar
[16]Hirata, M., Saussol, B. and Vaienti, S.. Statistics of return times: a general framework and new applications. Comm. Math. Phys. 206(1) (1999), 33–55.CrossRefGoogle Scholar
[17]Holland, M., Nicol, M. and Török, A.. Extreme value distributions for non-uniformly hyperbolic dynamical systems. Trans. Amer. Math. Soc. to appear.Google Scholar
[18]Le Cam, L.. An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10 (1960), 1181–1197.CrossRefGoogle Scholar
[19]Pène, F. and Saussol, B.. Back to balls in billiards. Comm. Math. Phys. 293(3) (2010), 837–866.CrossRefGoogle Scholar
[20]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.Google Scholar
[21]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar