Cookies

We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.


Durham Research Online
You are in:

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
Additional Information: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
Full text:PDF - Published Version (174Kb)
Full text:PDF - Published Version (230Kb)
Status:Peer-reviewed
Publisher Web site:http://www.dukeupress.edu/Catalog/ViewProduct.php?productid=45608
Publisher statement:2002 © Duke University Press
Record Created:08 Jun 2007
Last Modified:17 May 2010 15:00

Social bookmarking: del.icio.usConnoteaBibSonomyCiteULikeFacebookTwitterExport: EndNote, Zotero | BibTex
Usage statisticsLook up in GoogleScholar | Find in a UK Library