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16. Twist maps and Arnold diffusion for diffeomorphisms

From the book Variational Methods

  • Jean-Pierre Marco


We investigate a diffusion problem in the group Dκ of Cκ 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 Dκ with the uniform Cκ 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 Dκ.

We introduce the set Fκ ⊂ Dκ 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 Fκ 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 ∈ Fκ, there is a ball Bκ(f, ε) in Dκ such that for any g ∈ Bκ(f, ε), there is a ̃g ∈ Dκ−1, arbitrarily close to g in the Cκ−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].

© 2016 Walter de Gruyter GmbH, Berlin/Munich/Boston
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