Journal of Modern Dynamics (JMD)

Perfect retroreflectors and billiard dynamics

Pages: 33 - 48, Issue 1, January 2011      doi:10.3934/jmd.2011.5.33

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Pavel Bachurin - Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, United States (email)
Konstantin Khanin - Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario M5S 2E4, Canada (email)
Jens Marklof - School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom (email)
Alexander Plakhov - Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal (email)

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.

Keywords:  Recurrence, circle rotation, dynamical renormalization, homogeneous flow, billiards, retroreflectors.
Mathematics Subject Classification:  Primary: 37A50; Secondary: 37A17, 11K06.

Received: December 2009;      Revised: November 2010;      Available Online: April 2011.