Journal of Modern Dynamics (JMD)

No planar billiard possesses an open set of quadrilateral trajectories

Pages: 287 - 326, Issue 3, July 2012      doi:10.3934/jmd.2012.6.287

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Alexey Glutsyuk - CNRS, Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon 07, France (email)
Yury Kudryashov - National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russian Federation (email)

Abstract: The article is devoted to a particular case of Ivriĭ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise $C^4$-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.

Keywords:  Dynamical systems, billiards, periodic orbits.
Mathematics Subject Classification:  Primary: 58F22; Secondary: 34C25.

Received: January 2011;      Revised: May 2012;      Available Online: October 2012.