Topological Shadowing Methods in Arnold Diffusion: Weak Torsion and Multiple Time Scales

TL;DR

This paper proves the existence of Arnold diffusion in symplectic maps with normally hyperbolic invariant manifolds using topological shadowing, even under weak twist conditions and small separatrix splitting, with estimates on diffusion time.

Contribution

It generalizes previous methods by allowing non-uniform twist conditions and small separatrix splitting, providing a topological shadowing proof of Arnold diffusion.

Findings

Existence of Arnold diffusion under weak twist conditions

Diffusion occurs in directions with non-small splitting

Provides estimates for diffusion time

Abstract

Consider a symplectic map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave in [26], based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that the dynamics on the invariant manifold satisfy only a non-uniform twist condition, and, most importantly for applications, that the splitting of separatrices be small in certain directions and thus the associated drift in actions very slow; diffusion occurs in the directions of the manifold having non-small splitting. Furthermore we provide estimates for the diffusion time.