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 ɛ>0 from the flat torus 𝕋², 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 ∼τ_ɛ(ω), and respectively ∼R_ɛ(ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when 𝕋² is being identified with [0,1)². We prove that the probability measures on [0,∞) associated with the random variables ɛ∼τ_ɛ and ɛ∼R_ɛ are weakly convergent as ${{\varepsilon \rightarrow 0^+}}$ and explicitly compute the densities of the limits.