Global diffusion on a tight three-sphere

TL;DR

This paper demonstrates the existence of global diffusion trajectories on a tight three-sphere in a weakly coupled integrable Hamiltonian system, revealing complex dynamics and invariant structures.

Contribution

It establishes the persistence of a normally hyperbolic three-sphere and shows that trajectories can follow arbitrary invariant sets, demonstrating global diffusion in a tight contact structure.

Findings

Existence of trajectories following prescribed invariant tori and Aubry-Mather sets.

Persistence of the three-sphere as a normally hyperbolic invariant manifold under perturbation.

Global diffusion occurs on the tight three-sphere, indicating complex dynamical behavior.

Abstract

We consider an integrable Hamiltonian system weakly coupled with a pendulum-type system. For each energy level within some range, the uncoupled system is assumed to possess a normally hyperbolic invariant manifold diffeomorphic to a three-sphere, which bounds a strictly convex domain, and whose stable and unstable invariant manifolds coincide. The Hamiltonian flow on the three-sphere is equivalent to the Reeb flow for the induced contact form. The strict convexity condition implies that the contact structure on the three-sphere is tight. When a small, generic coupling is added to the system, the normally hyperbolic invariant manifold is preserved as a three-sphere, and the stable and unstable manifolds split, yielding transverse intersections. We show that there exist trajectories that follow any prescribed collection of invariant tori and Aubry-Mather sets within some global section of the flow restricted to the three-sphere. In this sense, we say that the perturbed system exhibits global diffusion on the tight three-sphere.