Topology of billiard problems, II

Farber, M.

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: Jun 8, 2007
• 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
• Public URL: https://durham-repository.worktribe.com/output/1599600
• Publisher URL: http://www.dukeupress.edu/Catalog/ViewProduct.php?productid=45608