In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in  to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards. In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In §5 we prove some estimates for the topological entropy of Lorentz gas.
All Science Journal Classification (ASJC) codes
- Applied Mathematics