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

Corentin Fierobe, Daniel Tsodikovich

TL;DR

The paper proves a local rigidity principle for the integrability of standard-like twist maps in the Frenkel-Kontorova setting, showing that a Suris potential $V_S$ with small eccentricity remains the only admissible integrable perturbation under $C^{r}$-small deformations when rational integrability on $[\tfrac{1}{6},\tfrac{1}{3}]$ is preserved. Central to the approach are action-angle coordinates adapted to Suris maps, a deformed Riesz basis in $L^2$ aligned with these coordinates, and precise action estimates that control how perturbations affect integrable structure. The results yield spectral rigidity statements: same action spectrum or same Mather beta function implies the perturbation is still a Suris potential, and any smooth isospectral deformation stays within the Suris class. Additionally, a periodic-rigidity statement excludes nontrivial short-period integrable perturbations, and the work shows higher-order integrability cannot occur within this framework. The methods parallel strategies used in billiards (e.g., ellipse) and contribute to a broader rigidity program for integrable twist maps.

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.

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

TL;DR

The paper proves a local rigidity principle for the integrability of standard-like twist maps in the Frenkel-Kontorova setting, showing that a Suris potential with small eccentricity remains the only admissible integrable perturbation under -small deformations when rational integrability on is preserved. Central to the approach are action-angle coordinates adapted to Suris maps, a deformed Riesz basis in aligned with these coordinates, and precise action estimates that control how perturbations affect integrable structure. The results yield spectral rigidity statements: same action spectrum or same Mather beta function implies the perturbation is still a Suris potential, and any smooth isospectral deformation stays within the Suris class. Additionally, a periodic-rigidity statement excludes nontrivial short-period integrable perturbations, and the work shows higher-order integrability cannot occur within this framework. The methods parallel strategies used in billiards (e.g., ellipse) and contribute to a broader rigidity program for integrable twist maps.

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 . 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.
Paper Structure (16 sections, 30 theorems, 228 equations)

This paper contains 16 sections, 30 theorems, 228 equations.

Table of Contents

  1. Introduction and Main Results
  2. Standard maps and Frenkel-Kontorova model
  3. Standard maps of Suris type
  4. Spectral rigidity
  5. Periodic rigidity
  6. Twist map of cylinders
  7. Action-angle coordinates for Suris map
  8. From Suris integrability to invariant curves
  9. Action angle coordinates
  10. An interesting special case
  11. The invariant curve of action $\frac{1}{4}$
  12. Deformed Riesz basis for $L^2$
  13. Action estimates
  14. Proof of Theorem \ref{['thm:LocalRigidity']}
  15. Proof of Corollaries \ref{['cor:SpecRigid1']} and \ref{['cor:SpecRigid2']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $a>0$, and $(V_{\varepsilon})_{0\leq \varepsilon<a}$ be a one-paremeter family of potentials $V_{\varepsilon}$, analytic in $\varepsilon$. Assume that for any $0 \leq \varepsilon < a$ and $V=V_{\varepsilon}$, the system equation:FK_equation admits a first integral $\Phi_{\varepsilon}$ of the for such that $\Phi_{\varepsilon}(x,x')=\Phi_{\varepsilon}(x',x)$ for any $0\leq \varepsilon < a$ and $

Theorems & Definitions (57)

  • Theorem 1: Suris suris1989integrable
  • Remark
  • Definition 1.1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • Definition 2.1
  • Proposition 2.1: Siburg, Theorem 1.3.7
  • Proposition 3.1: Suris
  • ...and 47 more