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

Michael Farber, Serge Tabachnikov

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)553-589
Number of pages37
JournalTopology
Volume41
Issue number3
DOIs
StatePublished - 2002

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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