Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

Abstract

The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric \(C^2\) strongly-convex domain \(D\) with boundary \(\partial D\), assume that the symplectic billiard map has a (simple) continuous invariant curve \(\delta \subset \mathcal{P}\) of rotation number \(1/4\) (winding once around \(\partial D\)) and consisting only of \(4\)-periodic orbits. If one of the parts between \(\delta\) and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then \(\partial D\) is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve \(\delta\), and the result is obtained by tracing back the problem directly to the totally integrable case.