Topology of billiard problems, II

Abstract

In this paper we give topological lower bounds on the number of periodic and of closed trajectories in strictly convex smooth billiards $T\subset \mathbf {R}\sp {m+1}$. Namely, for given $n$, we estimate the number of $n$-periodic billiard trajectories in $T$ and also estimate the number of billiard trajectories which start and end at a given point $A\in \partial T$ and make a prescribed number n of reflections at the boundary $\partial T$ of the billiard domain. We use variational reduction, admitting a finite group of symmetries, and apply a topological approach based on equivariant Morse and Lusternik-Schnirelman theories.