On conjugacy of convex billiards

TL;DR

This paper proves that if two convex billiard maps are smoothly conjugate near the boundary, then the domains are similar, and explores implications for the Birkhoff conjecture and spectral geometry.

Contribution

It establishes a conjugacy rigidity result for convex billiards and connects it to the Birkhoff conjecture and spectral geometry questions.

Findings

Conjugate billiard maps imply similar domains via rescaling and isometry.

Provides a conditional proof related to the Birkhoff conjecture.

Links spectral data to geometric domain equivalence.

Abstract

Given a strictly convex domain $\Omega$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we answer a relatively old question of Guillemin. We show that if two billiard maps are $C^{1,\alpha}$-conjugate near the boundary, for some $\alpha > 1/2$, then the corresponding domains are similar, i.e. they can be obtained one from the other by a rescaling and an isometry. As an application, we prove a conditional version of Birkhoff conjecture on the integrability of planar billiards and show that the original conjecture is equivalent to what we call an "Extension problem". Quite interestingly, our result and a positive solution to this extension problem would provide an answer to a closely related question in spectral theory: if the marked length spectra of two domains are the same, is it true that they are isometric?