#### Vol. 295, No. 2, 2018

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Nonsmooth convex caustics for Birkhoff billiards

### Maxim Arnold and Misha Bialy

Vol. 295 (2018), No. 2, 257–269
##### 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.

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##### 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
##### Authors
 Maxim Arnold Department of Mathematical Sciences University of Texas at Dallas Richardson, TX United States Misha Bialy Tel Aviv University Tel Aviv Israel