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 ∂ D , assume that the symplectic billiard map has a (simple) continuous invariant curve δ ⊂ P of rotation number 1 / 4 (winding once around ∂ D ) and consisting only of 4-periodic orbits. If one of the parts between δ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then ∂ 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 δ, and the result is obtained by tracing back the problem directly to the totally integrable case.