Deformational spectral rigidity of axially symmetric symplectic billiards

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.