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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics