We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in Rm+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 Sm, i.e., the space of n-tuples of points (x1,...,xn), where xi ∈ Sm and xi ≠ xi+1 for i = 1,...,n.
All Science Journal Classification (ASJC) codes
- Geometry and Topology