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 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.

Authors

Jacopo De Simoi

Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON Canada M5S 2E4

Vadim Kaloshin

Department of Mathematics, University of Maryland, College Park, College Park, MD 20742

Qiaoling Wei

Department of Mathematics, Capital Normal University, Beijing 100048, PR China, and University of Maryland, College Park, College Park, MD 20742

Hamid Hezari

Department of Mathematics, University of California, Irvine, Irvine, CA 92697