Dynamical Spectral rigidity among $\mathbb Z_2$-symmetric strictly convex domains close to a circle

TL;DR

This paper proves that certain symmetric convex domains close to a circle are uniquely determined by their billiard spectral data, meaning no non-trivial shape deformations preserve all periodic orbit lengths.

Contribution

It establishes dynamical spectral rigidity for smooth, symmetric convex domains near a circle, partially answering Sarnak's question.

Findings

Spectral rigidity holds for domains close to a circle with $ ext{Z}_2$ symmetry.

Deformations preserving all periodic orbit lengths are necessarily isometric.

Results apply to a class of smooth, convex billiard domains.

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 which 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.