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

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

1999-11-28

arxiv.org/abs/math/9911226v1

Mathematics

Differential Geometry

38 pages, 4 figures

Scientific paper

We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. 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 $(x_1, ..., x_n)$, where $x_i\in S^m$ and $x_i\ne x_{i+1}$ for i=1,2, ..., n.

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