Deformational spectral rigidity of axially symmetric symplectic billiards

Abstract Symplectic billiards are discrete dynamical systems which were introduced by Albers and Tabachnikov and take place in a strongly convex bounded planar domain with smooth boundary. They are described by the symplectic law of reflection , in contrast to the elastic reflection law of Birkhoff billiards. In this paper, we prove a version of dynamical spectral rigidity for symplectic billiards which is a counterpart to previous results on classical billiards by De Simoi, Kaloshin and Wei. Namely, we show that close to an ellipse, any sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area spectra or is trivial, in the sense that the domains differ by area-preserving affine transformations of the plane. We also prove that in general—that is, even if the domains are not close to an ellipse—any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite-dimensional space.