Lower bounds on the number of closed trajectories of generalized billiards

Abstract

Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from the boundary making the angle of incidence equal the angle of reflection. The billiard motion can happen to be a closed (or periodic) one when the billiard ball rebounds k times and then gets to the initial position with the same speed vector as in the beginning. The study of closed billiard trajectories is due to George Birkhoff who in 1927 proved a lower estimate for the number of closed billiard trajectories of a certain period k. We consider the most general case when the billiard ball reflects from an arbitrary submanifold of a Euclidean space. We prove Morse inequalities in this situation and apply them to find a lower estimate for the number of closed billiard trajectories of any prime period in terms of Betti numbers of the given manifold.