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

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 & epsilon;)& epsilon;& ISIN;R, obtained by perturbing the generating function of an analytic map f by a family of potentials {& epsilon;G}& epsilon;& ISIN;R. Firstly, for an analytic G and for (m, n) & ISIN; Zd x N* with m and n coprime, we investigate the topological structure of the set of & epsilon; & ISIN; R for which f & epsilon; 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 prop erties of Lagrangian periodic tori, which we believe is of its own interest. & COPY; 2023 Elsevier Inc. All rights reserved.