Ergodicity and annular homeomorphisms of the torus

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.

14 pages, 3 figures. This version contains substantial improvements to the exposition

References in corpus (5)

• Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion
• Strictly Toral Dynamics
• Dynamics of homeomorphisms of the torus homotopic to Dehn twists
• Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-Torus
• On torus homeomorphisms whose rotation set is an interval

Cited by in corpus (2)

• Bounded and unbounded behavior for area-preserving rational pseudo-rotations
• A characterization of annularity for area-preserving toral homeomorphisms