Diffusion along transition chains of invariant tori and Aubry-Mather sets

TL;DR

This paper presents a topological mechanism for the existence of diffusing orbits in Hamiltonian systems, utilizing transition chains of invariant tori and Aubry-Mather sets, advancing understanding of Arnold diffusion.

Contribution

It introduces a topological, constructive method to prove the existence of diffusing trajectories following transition chains and Aubry-Mather sets in Hamiltonian systems.

Findings

Existence of diffusing trajectories following transition chains.

Construction of trajectories crossing gaps and following Aubry-Mather sets.

Application to the large gap problem in Hamiltonian systems.

Abstract

We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map, (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains, i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence, (iv) the transition chains of tori are interspersed with gaps created by resonances, (v) within each gap there is prescribed a finite collection of Aubry-Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.