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

2022/02/01 by Arnaud, Marie-Claude, Massetti, Jessica Elisa, Sorrentino, Alfonso · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2202.00313

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_ε)_{ε∈ ℝ}, obtained by perturbing the generating function of an analytic map f by a family of potentials \ε G\_{ε∈ ℝ}. Firstly, for an analytic G and for (m,n)∈ ℤ× ℕ^* with m and n coprime, we investigate the topological structure of the set of ε∈ ℝ for which f_ε 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.

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