Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle (Appendix B coauthored with H. Hezari)

Abstract

We show that any sufficiently (finitely) smooth $\mathbb{Z}_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.