Nearly circular domains which are integrable close to the boundary are ellipses

TL;DR

This paper proves that nearly circular domains with integrability close to the boundary must be ellipses, extending previous results and linking local boundary integrability to global integrability in billiard dynamics.

Contribution

It establishes a local version of the Birkhoff conjecture for nearly circular domains, showing such domains are necessarily ellipses if integrability is preserved near the boundary.

Findings

Small perturbations of ellipses with near-boundary integrability are themselves ellipses.

Integrability near the boundary implies global integrability for nearly circular domains.

Higher order conditions for preserving integrable rational caustics are derived and analyzed.

Abstract

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in [1], where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.