On the local Birkhoff conjecture for convex billiards

2018/06/15 by Vadim Kaloshin, Alfonso Sorrentino · 1 citation
Physics and Astronomy · Mathematics · #Quantum chaos and dynamical systems #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology

paper · doi:10.4007/annals.2018.188.1.6

openalex publication_date 2018/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.

Citations

• The propagation of singularities along gliding rays
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