No planar billiard possesses an open set of quadrilateral trajectories

Alexey Glutsyuk; Yury Kudryashov

doi:10.3934/jmd.2012.6.287

Article Contents

2012, Volume 6, Issue 3: 287-326. Doi: 10.3934/jmd.2012.6.287

This issue Previous Article Next Article The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$

No planar billiard possesses an open set of quadrilateral trajectories

Alexey Glutsyuk^1, and
Yury Kudryashov^2,

1.
CNRS, Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon 07
2.
National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000

Received: December 31, 2010

Revised: April 30, 2012

Published: September 2012

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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.

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Mathematics Subject Classification: Primary: 58F22; Secondary: 34C25.

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