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

Abstract

We consider a billiard in the punctured torus obtained by removing a small disk of radius ɛ>0 from the flat torus 𝕋2, 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 ˜τɛ(ω), and respectively ˜Rɛ(ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when 𝕋2 is being identified with [0,1)2. We prove that the probability measures on [0,∞) associated with the random variables ɛ˜τɛ and ɛ˜Rɛ are weakly convergent as \({{\varepsilon \rightarrow 0^+}}\) and explicitly compute the densities of the limits.

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Correspondence to Florin P. Boca.

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Research partially supported by ANSTI grant C6189/2000.

Communicated by G. Gallavotti

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Boca, F., Gologan, R. & Zaharescu, A. The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus. Commun. Math. Phys. 240, 53–73 (2003). https://doi.org/10.1007/s00220-003-0907-4

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Keywords

  • Phase Space
  • Probability Measure
  • Exit Time
  • Small Disk
  • Puncture Torus