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
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 -