The statistics of the trajectory in a certain billiard in a flat two-torus

Abstract

We consider a billiard in the punctured torus obtained by removing a small disk from the two-dimensional flat torus, with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. We prove that the probability measures on $[0,\infty)$ associated with the first exit time are weakly convergent when the size of the puncture tends to zero and explicitly compute the density of the limit.