16. Twist maps and Arnold diffusion for diffeomorphisms

Abstract

We investigate a diffusion problem in the group D^κ of C^κ symplectic diffeomorphisms of the annulus A²,with compact support contained in T²×I²,where I ⊂ ℝ is a fixed interval containing [0, 1] in its interior. We endow D^κ with the uniform C^κ topology.

Let (θ, r) be the usual angle-action coordinates on A². We say that a diffeomorphism inDκ satisfies the diffusion propertywhen it admits orbitswhose action r1 starts close to 0 and later gets close to 1. Our problem is to describe the occurrence of the diffusion property in D^κ.

We introduce the set F^κ ⊂ D^κ of diffeomorphisms of the form f(x₁, x₂) = (f₁(x₁), f₂(x₂)). The restriction of f₁ to T × [0, 1] is a symplectic twist map, which leaves the circles T×{0} and T×{1} invariantwith Diophantine rotation, while f₂ is a symplectic diffeomorphism that admits a hyperbolic fixed point O₂ (whose expansion and contraction dominate those of f₁) together with a transverse homoclinic point P₂. For κ large enough, a diffeomorphism in F_κ does not satisfy the diffusion property and so, by analogy with the Hamiltonian setting for Arnold diffusion, it can be legitimately considered as an “unperturbed” system.

Our main result, in the spirit of Arnold’s questions, is the following: when κ is large enough, given f ∈ F^κ, there is a ball B^κ(f, ε) in D^κ such that for any g ∈ B^κ(f, ε), there is a ̃g ∈ D^κ−1, arbitrarily close to g in the C^κ−1 topology, which satisfies the diffusion property.

The diffeomorphisms we deal with here can be seen as models for Poincare sections of near-integrable convex Hamiltonian systems in the neighborhood of double resonances, following the approach of [27].