Nonsmooth convex caustics for Birkhoff billiards

Maxim Arnold, Misha Bialy

Research output: Contribution to journalArticlepeer-review


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
Issue number2
StatePublished - 2018


  • Birkhoff billiard
  • Convex caustics
  • String construction


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