Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

Abstract

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R m+1 for m⩾3. For plane billiards (when m=1) such bounds were obtained by Birkhoff in the 1920s. Our proof is based on topological methods of calculus of variations — equivariant Morse and Lusternik–Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere S m , i.e., the space of n-tuples of points ( x 1,…, x n ), where x i ∈ S m and x i ≠ x i+1 for i=1,…, n.