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

1999/11/28 by Michael Farber, Farber, Michael, Serge Tabachnikov +1
Mathematics · #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #math.AT #math.DG

paper · pdf · doi:10.48550/arxiv.math/9911226

38 pages, 4 figures

arxiv created 1999/11/28 · arxiv updated 2009/11/30

Abstract

We give lowed 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 G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik - Schirelman 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 (x₁, ..., x_n), where x_i∈ S^m and x_i≠ x_i+1 for i=1,2, ..., n.

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