A billiard table close to an ellipse is deformationally spectrally rigid among dihedrally symmetric domains

Abstract

We prove that a a strongly convex planar domain (Birkhoff table) with dihedral symmetry, which is sufficiently close in a finitely smooth topology to an ellipse, is deformationally spectrally rigid within the class of domains preserving this symmetry. More precisely, any smooth one-parameter family of such domains that preserves the length spectrum (i.e., the set of lengths of periodic billiard orbits) must consist only of rigid motions of the initial domain. The proof combines two types of dynamical data: the asymptotic behavior of certain symmetric periodic orbits, as previously used in the rigidity of nearly circular domains, and new spectral information derived from KAM invariant curves, obtained from Mather's beta function and its derivatives (in the Whitney sense) at some suitable rotation numbers.