On the fragility of periodic tori for families of symplectic twist maps

TL;DR

This paper investigates the fragility of Lagrangian periodic tori in symplectic twist maps, showing that such tori are highly sensitive to perturbations and establishing a rigidity result for integrable cases.

Contribution

It provides a detailed analysis of the topological and dynamical structure of periodic tori under perturbations, revealing finiteness properties and rigidity phenomena in symplectic twist maps.

Findings

The set of parameters with Lagrangian periodic tori is finite under certain conditions.

Rigidity results for integrable symplectic twist maps are established.

The analysis applies in any dimension and enhances understanding of the stability of invariant tori.

Abstract

In this article we study the fragility of Lagrangian periodic tori for symplectic twist maps of the $2d$-dimensional annulus and prove a rigidity result for completely integrable ones. More specifically, we consider $1$-parameter families of symplectic twist maps $(f_\varepsilon)_{\varepsilon\in \mathbb{R}}$, obtained by perturbing the generating function of an analytic map $f$ by a family of potentials $\{\varepsilon G\}_{\varepsilon\in \mathbb{R}}$. Firstly, for an analytic $G$ and for $(m,n)\in \mathbb{Z}\times \mathbb{N}^*$ with $m$ and $n$ coprime, we investigate the topological structure of the set of $\varepsilon\in \mathbb{R}$ for which $f_\varepsilon$ admits a Lagrangian periodic torus of rotation vector $(m,n)$. In particular we prove that, under a suitable non-degeneracy condition on $f$, this set consists of at most finitely many points. Then, we exploit this to deduce a rigidity result for integrable symplectic twist maps, in the case of deformations produced by a $C^2$ potential. Our analysis, which holds in any dimension, is based on a thorough investigation of the geometric and dynamical properties of Lagrangian periodic tori, which we believe is of its own interest.