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Topology of billiard problems, II

Farber, M.

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Authors

M. Farber



Abstract

Part I. Let T⊂Rm+1T⊂Rm+1 be a strictly convex domain bounded by a smooth hypersurface X=\partialTX=\partialT. In this paper we find lower bounds on the number of billiard trajectories in TT which have a prescribed initial point A∈XA∈X, a prescribed final point B∈XB∈X, and make a prescribed number nn of reflections at the boundary XX. We apply a topological approach based on the calculation of cohomology rings of certain configuration spaces of SmSm. Part I. In this paper we give topological lower bounds on the number of periodic and of closed trajectories in strictly convex smooth billiards T⊂Rm+1T⊂Rm+1. Namely, for given nn, we estimate the number of nn-periodic billiard trajectories in TT and also estimate the number of billiard trajectories which start and end at a given point A∈∂TA∈∂T and make a prescribed number n of reflections at the boundary ∂T∂T of the billiard domain. We use variational reduction, admitting a finite group of symmetries, and apply a topological approach based on equivariant Morse and Lusternik-Schnirelman theories.

Citation

Farber, M. (2002). Topology of billiard problems, II. Duke Mathematical Journal, 115(3), 559-621. https://doi.org/10.1215/s0012-7094-02-11535-x%2C+10.1215/s0012-7094-02-11536-1

Journal Article Type Article
Publication Date 2002
Deposit Date Jun 8, 2007
Publicly Available Date Mar 28, 2024
Journal Duke Mathematical Journal
Print ISSN 0012-7094
Electronic ISSN 1547-7398
Publisher Duke University Press
Peer Reviewed Peer Reviewed
Volume 115
Issue 3
Pages 559-621
DOI https://doi.org/10.1215/s0012-7094-02-11535-x%2C+10.1215/s0012-7094-02-11536-1
Publisher URL http://www.dukeupress.edu/Catalog/ViewProduct.php?productid=45608

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Copyright Statement
2002 © Duke University Press





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