Nonsmooth convex caustics for Birkhoff billiards

Maxim Arnold and Misha Bialy

Vol. 295 (2018), No. 2, 257–269

DOI: 10.2140/pjm.2018.295.257

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 $C^2$-smooth convex billiard tables with a nonsmooth convex caustic.

Keywords

string construction, convex caustics, Birkhoff billiard

Mathematical Subject Classification 2010

Primary: 37E30, 37E40

Secondary: 78A05

Milestones

Received: 14 August 2017

Revised: 29 December 2017

Accepted: 22 January 2018

Published: 11 April 2018