Perfect retroreflectors and billiard dynamics

Pavel Bachurin; Konstantin Khanin; Jens Marklof; Alexander Plakhov

doi:10.3934/jmd.2011.5.33

Article Contents

2011, Volume 5, Issue 1: 33-48. Doi: 10.3934/jmd.2011.5.33

This issue Previous Article Shimura and Teichmüller curves Next Article Boundary unitary representations-irreducibility and rigidity

Perfect retroreflectors and billiard dynamics

1.
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
2.
Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario M5S 2E4
3.
School of Mathematics, University of Bristol, Bristol BS8 1TW
4.
Department of Mathematics, University of Aveiro, Aveiro 3810-193

Received: December 2009

Revised: November 2010

Published: April 2011

Abstract / Introduction Related Papers Cited by

Abstract

We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.

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Mathematics Subject Classification: Primary: 37A50; Secondary: 37A17, 11K06.

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