Nonsmooth convex caustics for Birkhoff billiards

Maxim Arnold, Misha Bialy

Research output: Contribution to journalArticlepeer-review

Abstract

This paper is devoted to the examination of the properties of the string construction for the Birkhoff billiard. Based on purely geometric considerations, string construction is suited to providing a table for the Birkhoff billiard, having the prescribed caustic. Exploiting this framework together with the properties of convex caustics, we give a geometric proof of a result by Innami first proved in 2002 by means of Aubry-Mather theory. In the second part of the paper we show that applying the string construction one can find a new collection of examples of C2-smooth convex billiard tables with a nonsmooth convex caustic.

Original languageEnglish
Pages (from-to)257-269
Number of pages13
JournalPacific Journal of Mathematics
Volume295
Issue number2
DOIs
StatePublished - 2018

Keywords

  • Birkhoff billiard
  • Convex caustics
  • String construction

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