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

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 τ̃_ε(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 ε̃_ε and ε R̃ _ε are weakly convergent as ε → 0⁺ and explicitly compute the densities of the limits.