Local rigidity for symplectic billiards

TL;DR

This paper proves that if a domain near an ellipse has a symplectic billiard map that is rationally integrable, then the domain must be an ellipse, establishing a local rigidity result similar to classical billiard results.

Contribution

It establishes a local rigidity theorem for symplectic billiards, showing that near an ellipse, integrability implies the domain is an ellipse, extending classical billiard rigidity results.

Findings

Domains close to ellipses with rationally integrable symplectic billiards are ellipses.

The result parallels classical billiard rigidity theorems by Avila, De Simoi, and Kaloshin.

Provides a new rigidity criterion for symplectic billiard systems.

Abstract

We show a local rigidity result for the integrability of symplectic billiards. We prove that any domain which is close to an ellipse, and for which the symplectic billiard map is rationally integrable must be an ellipse as well. This is in spirit of the result of Avila, De Simoi, and Kaloshin for Birkhoff billiards.