On the persistence of periodic tori for symplectic twist maps and the rigidity of integrable twist maps

Abstract

In this article we study the persistence 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 pfεq εPR , obtained by perturbing the generating function of an analytic map f by a family of potentials tεGu εPR . Firstly, for an analytic G and for pm, nq P Z ˆN˚w ith m and n coprime, we investigate the topological structure of the set of ε P R for which fε admits a Lagrangian periodic torus of rotation vector pm, nq. 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.

Deformational rigidity of integrable metrics on the torus

“…Moreover, the perturbation λ depends solely on the angular but not the action coordinates of the unperturbed problem (see Theorem 1.6). Although the analog of this result for symplectic twist maps in this peculiar setting has already been shown in [5,74] by methods similar to ours, we reprove it by pursuing an only slightly different but original strategy, which is suitable for certain inevitable modifications for the proofs of the more general statements under item (ii) and (iii). These two cases (corresponding to surfaces of revolution and general Liouville metrics, see Section 3) build on perturbative estimates for (possibly infinitely many) systems of linear equations for the Fourier coefficients.…”

“…In a more recent work, Arnaud-Massetti-Sorrentino [5] (replacing the earlier preprint [74]) studied the rigidity of integrable symplectic twist maps on the 2d-dimensional annulus T d × R d . More precisely, they consider one-parameter families (f ε ) ε∈R of symplectic twist maps f ε (x, p) = f 0 (x, p + ε∇G(x)) and prove two main rigidity results: First, in the analytic category for f 0 and the perturbation G, if a single rational invariant Lagrangian graph of f ε exists for infinitely many values of ε (e.g.…”

“…Note that in this second result, the entire phase space is foliated by invariant tori, and the perturbation solely depends on the angle variables of the dynamical system. In this sense, Theorem I can -morally -be viewed as a special case of the second result in [5] (see also [74,Theorem 2]), but Theorem II and Theorem III generalize this statement to more general functional dependencies of the perturbation. Apart from this, our general results (i.e.…”