Strictly Toral Dynamics

TL;DR

This paper studies the dynamics of nonwandering, area-preserving homeomorphisms of the torus homotopic to the identity, revealing a structured decomposition into elliptic islands and chaotic regions with bounded islands and rich rotational behavior.

Contribution

It introduces a detailed classification of points into inessential and essential sets, proving boundedness of elliptic islands and extending previous descriptions of toral dynamics.

Findings

Inessential points form disjoint periodic disks.

Essential points constitute a complex, chaotic continuum.

Elliptic islands are bounded, enabling sharp diffusion bounds.

Abstract

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points $ine(f)$ is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points $ess(f)$ is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by J\"ager. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.