The average length of a trajectory in a certain billiard in a flat two-torus

F. P. Boca, R. N. Gologan, A. Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

We remove a small disc of radius ε > 0 from the flat torus T2 and consider a point-like particle that starts moving from the center of the disk with linear trajectory under angle ω. Let ~τ(ε)(ω) denote the first exit time of the particle. For any interval I . [0, 2π), any r > 0, and any δ > 0, we estimate the moments of ~τ on I and prove the asymptotic formula (equation presented), where cr is the constant (equation presented). A similar estimate is obtained for the moments of the number of reflections in the side cushions when T2 is identified with [0, 1)2.

Original languageEnglish (US)
Pages (from-to)303-330
Number of pages28
JournalNew York Journal of Mathematics
Volume9
StatePublished - Jan 1 2003

ASJC Scopus subject areas

  • Mathematics(all)

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