Duke Mathematical Journal

Topology of billiard problems, II

Michael Farber

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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.

Article information

Duke Math. J., Volume 115, Number 3 (2002), 587-621.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R80: Discriminantal varieties, configuration spaces
Secondary: 37C25: Fixed points, periodic points, fixed-point index theory 37D50: Hyperbolic systems with singularities (billiards, etc.) 37J10: Symplectic mappings, fixed points 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Farber, Michael. Topology of billiard problems, II. Duke Math. J. 115 (2002), no. 3, 587--621. doi:10.1215/S0012-7094-02-11536-1. https://projecteuclid.org/euclid.dmj/1085598180

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See also

  • See also: Michael Farber. Topology of billiard problems, I. Duke Math. J. Vol. 115, No. 3 (2003), pp. 559-585.