Lower bounds for the number of closed billiard trajectories of period 2and 3in manifolds embedded in Euclidean space

Abstract

Let T ⊂ ℝ2 be a smooth strictly convex domain bounded by a smooth curve M = ∂ T. The billiard ball is a point which moves in T along a straight line and rebounds from M making the angle of incidence equal to the angle of reflection. The classical problem by G. Birkhoff is to find the lower estimate for the number of closed billiard trajectories with p reflection points. In this paper, we give a definition of a periodic billiard trajectory in a smooth closed m-dimensional manifold M ⊂ ℝn, find a lower bound for the number of 3-periodic billiard trajectories, and give a new proof of P. Pushkar's estimate for 2-periodic trajectories.