Rigidity of the Suris' potential in the Frenkel-Kontorova Model

Abstract

The goal of this paper is to establish a local rigidity result for the integrability of standard-like maps. The main focus of the paper is the remarkable integrable potential discovered by Suris in the 80's. We show that locally, the integrability of this potential is rigid. The proof relies on a similar strategy that was used for billiards in an ellipse, and involves developing the action-angle coordinates for this system, and exploiting it to construct a Riesz basis for $L^2$. As a corollary, we obtain a spectral rigidity result for this setting. Finally, we study the integrability question in the setting of potentials that are periodic.