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

Michael Farber, Sergei Tabachnikov

Research output: Contribution to journalArticle

  • 19 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.

LanguageEnglish (US)
Pages553-589
Number of pages37
JournalTopology
Volume41
Issue number3
DOIs
StatePublished - Feb 2 2002

Fingerprint

Periodic Trajectories
Billiards
Configuration Space
Topology
Topological Methods
Equivariant Cohomology
Cohomology Ring
Calculus of variations
Strictly Convex
Equivariant
Lower bound

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

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Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards. / Farber, Michael; Tabachnikov, Sergei.

In: Topology, Vol. 41, No. 3, 02.02.2002, p. 553-589.

Research output: Contribution to journalArticle

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