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

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 (fε)ε∈R, obtained by perturbing the generating function of an analytic map f by a family of potentials {εG}ε∈R. Firstly, for an analytic G and for (m,n)∈Zd×N⁎ with m and n coprime, we investigate the topological structure of the set of ε∈R for which fε 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 C2 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.