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# Topology of billiard problems : I and II.

Farber, M. (2002) 'Topology of billiard problems : I and II.', Duke mathematical journal., 115 (3). pp. 559-621.

## Abstract

Part I. Let $T\subset \mathbf {R}\sp {m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partialT$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed initial point $A\in X$, a prescribed final point $B\in X$, and make a prescribed number $n$ of reflections at the boundary $X$. We apply a topological approach based on the calculation of cohomology rings of certain configuration spaces of $S\sp m$. Part I. In this paper we give topological lower bounds on the number of periodic and of closed trajectories in strictly convex smooth billiards $T\subset \mathbf {R}\sp {m+1}$. Namely, for given $n$, we estimate the number of $n$-periodic billiard trajectories in $T$ and also estimate the number of billiard trajectories which start and end at a given point $A\in \partial T$ and make a prescribed number n of reflections at the boundary $\partial T$ 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.

Item Type: Article Part I : http://dx.doi.org/10.1215/S0012-7094-02-11535-X, Part II : http://dx.doi.org/10.1215/S0012-7094-02-11536-1 PDF - Published Version (174Kb) PDF - Published Version (230Kb) Peer-reviewed http://www.dukeupress.edu/Catalog/ViewProduct.php?productid=45608 2002 © Duke University Press 08 Jun 2007 17 May 2010 15:00

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