TY - JOUR

T1 - The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

AU - Boca, Florin P.

AU - Gologan, Radu N.

AU - Zaharescu, Alexandru

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2003/9

Y1 - 2003/9

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-003-0907-4

DO - 10.1007/s00220-003-0907-4

M3 - Article

AN - SCOPUS:0242473182

VL - 240

SP - 53

EP - 73

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1-2

ER -