The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

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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 τ̃ε(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 sign2 is being identified with [0, 1) 2. We prove that the probability measures on (0, ∞) associated with the random variables ε̃ε and ε R̃ ε are weakly convergent as ε → 0+ and explicitly compute the densities of the limits.

Original languageEnglish (US)
Pages (from-to)53-73
Number of pages21
JournalCommunications in Mathematical Physics
Volume240
Issue number1-2
DOIs
StatePublished - Sep 2003

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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