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Back to Balls in Billiards

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Abstract

We consider a billiard in the plane with periodic configuration of convex scatterers. This system is recurrent, in the sense that almost every orbit comes back arbitrarily close to the initial point. In this paper we study the time needed to get back in an ε-ball about the initial point, in the phase space and also for the position, in the limit when ε → 0. We establish the existence of an almost sure convergence rate, and prove a convergence in distribution for the rescaled return times.

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Authors and Affiliations

  1. Université Européenne de Bretagne, Brest, France

    Françoise Pène & Benoît Saussol

  2. Université de Brest, Laboratoire de Mathématiques, CNRS UMR 6205, 6 avenue Victor Le Gorgeu, F-29238, Brest, France

    Françoise Pène & Benoît Saussol

Authors
  1. Françoise Pène
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  2. Benoît Saussol
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Corresponding author

Correspondence to Benoît Saussol.

Additional information

Communicated by G. Gallavotti

Françoise Pène is partially supported by the ANR project TEMI (Théorie Ergodique en mesure infinie).

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Pène, F., Saussol, B. Back to Balls in Billiards. Commun. Math. Phys. 293, 837–866 (2010). https://doi.org/10.1007/s00220-009-0911-4

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  • DOI: https://doi.org/10.1007/s00220-009-0911-4

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