Coexistence of periodic-2 and periodic-3 caustics for nearly circular analytic billiard maps

TL;DR

This paper investigates conditions under which nearly circular analytic billiard maps can maintain coexistence of periodic 2 and 3 caustics, concluding that only isometric deformations preserve this coexistence.

Contribution

It proves that for symmetric analytic deformations of the circle, the coexistence of periodic 2 and 3 caustics implies the deformation must be an isometry.

Findings

Coexistence of periodic 2 and 3 caustics requires isometric deformation.

Symmetric analytic deformations with certain Fourier decay cannot preserve caustic coexistence unless isometric.

The result characterizes the rigidity of caustic coexistence under analytic deformations.

Abstract

For symmetrically analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of periodic $2,3$ caustics is that the deformation has to be an isometric transformation.