## Abstract

We consider a billiard in the punctured torus obtained by removing a small disk of radius ε > 0 from the flat torus double-struck T sign ^{2}, with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity ω only. Let τ̃_{ε}(ω), and respectively R̃ _{ε}(ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when double-struck T sign^{2} is being identified with [0, 1) ^{2}. We prove that the probability measures on (0, ∞) associated with the random variables ε̃_{ε} and ε R̃ _{ε} are weakly convergent as ε → 0^{+} and explicitly compute the densities of the limits.

Original language | English (US) |
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Pages (from-to) | 53-73 |

Number of pages | 21 |

Journal | Communications in Mathematical Physics |

Volume | 240 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 2003 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics