Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric C² strongly-convex domain D with boundary D, assume that the symplectic billiard map has a (simple) continuous invariant curve P of rotation number 1/4 (winding once around D) and consisting only of 4-periodic orbits. If one of the parts between 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, and the result is obtained by tracing back the problem directly to the totally integrable case.