Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma

Abstract

We consider di erentiable maps having an invariant torus with normal behavior having a central part. We p r o ve the existence and regularity of pseudostable manifolds and regularity with respect to parameters. Then we p r o ve a l a m bda lemma in this setting for C 2 maps.

Arnold diffusion for a complete family of perturbations

“…Using the symmetry of H r , introducing I i = −I i for i < 0, we have the pseudo-orbit {(I i , θ i ), |i| ≤ N } ⊂ H r . Using standard shadowing results in [FM00,FM03] (changing slightly I i to obtain an irrational frequency of the inner map, if necessary ) or newer results like the corollary 3.5 of [GLS14], there exists a trajectory of the system such that for some T , I 0 ≤ −I * and I(T ) ≥ I * . If a 10 < 0, changing H r to H l all the previous reasoning applies.…”

A λ-lemma for normally hyperbolic invariant manifolds

“…This follows by applying the λ-lemma for partially hyperbolic tori of Fontich and Martin [12] to C 0 and C 1 . It follows from this lemma that for all ǫ > 0, there exists a positive integer K ′ such that for all…”

“…Theorem 4 implies that the λ-lemma proven in Theorem 3 applies to the three degreeof-freedom Hamiltonian systems defined by (12).…”

Arnold diffusion of the discrete nonlinear Schrödinger equation

“…After enough iterations of F , one can obtain some estimate of closeness to W u . The novelty of [14] is that they also track the tangent space at every point in S and off W s . This is necessary in order to obtain the claim of the lemma.…”