Topology of billiard problems, II

In this paper we give topological lower bounds on the number of periodic and of closed trajectories in strictly convex smooth billiards T ⊂ R^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 ∈ ∂T and make a prescribed number n of reflections at the boundary ∂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.