ON THE DISCONTINUITIES OF THE BOUNDARY IN BILLIARDS

This work concerns billiards in domains of R2. The set S of pre-images of discontinuities of T, the bouncing map induced by the flow, is studied, particularly those discontinuities due to points where the boundary is not C1. We prove that if the angle at the discontinuity is a divisor of pi there exists a power of T which is continuous around such a point, and that under conditions of symmetry this power is locally differentiable. It is assumed that continuity may often be extended to certain areas of phase space. This property is used to show by means of numerical simulations of two billiards how S fashions the phase space. In general if the angle is not a divisor of pi, S separates the distinct areas of quasi-integrable motion. One of the billiards is isomorphic to a mechanical model of three interacting particles on a line.