Ergodicity and annular homeomorphisms of the torus

TL;DR

This paper investigates the dynamics of certain torus homeomorphisms, showing that ergodic maps with specific rotation set properties must have some power that is an annular homeomorphism.

Contribution

It establishes a link between ergodicity, rotation set geometry, and annular behavior for torus homeomorphisms homotopic to the identity.

Findings

If the rotation set is a rational-slope line segment containing a rational point

And the average rotation vector is irrational, then some iterate of the map is annular

The result connects ergodic properties with topological dynamics of the map.

Abstract

Let $f: \mathbb{T}^2 \to \mathbb{T}^2$ be a homeomorphism homotopic to the identity and $F: \mathbb{R}^2 \to \mathbb{R}^2$ a lift of $f$ such that the rotation set $\rho(F)$ is a line segment of rational slope containing a point in $\mathbb{Q}^2$. We prove that if $f$ is ergodic with respect to the Lebesgue measure on the torus and the average rotation vector (with respect to same measure) does not belong to $\mathbb{Q}^2$ then some power of $f$ is an annular homeomorphism.