Non-smooth convex caustics for Birkhoff billiard

TL;DR

This paper explores the geometric properties of string constructions in Birkhoff billiards, providing a new proof of a known result and discovering new examples of billiard tables with non-smooth convex caustics.

Contribution

It offers a geometric proof of Innami's 2002 result and introduces new convex billiard tables with non-smooth caustics using string construction methods.

Findings

Geometric proof of Innami's result from 2002

Construction of new convex billiard tables with non-smooth caustics

Application of string construction to generate examples

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 provide 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 non-smooth convex caustic.