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.

Arnaud, M., Massetti, J., Sorrentino, A. (2023). On the fragility of periodic tori for families of symplectic twist maps. ADVANCES IN MATHEMATICS, 429 [10.1016/j.aim.2023.109175].

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

Massetti, JE;Sorrentino, A
2023-01-01

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 & 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.
2023
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
English
Symplectic twist maps
Periodic tori
Integrability
Arnaud, M., Massetti, J., Sorrentino, A. (2023). On the fragility of periodic tori for families of symplectic twist maps. ADVANCES IN MATHEMATICS, 429 [10.1016/j.aim.2023.109175].
Arnaud, M; Massetti, J; Sorrentino, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/336603
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