Irrational rotation factors for conservative torus homeomorphisms

TL;DR

This paper characterizes when conservative torus homeomorphisms have irrational rotation factors, linking the existence of such factors to bounded deviations and a well-defined rotation speed in some rational direction.

Contribution

It provides an equivalent condition for the existence of irrational rotation factors in non-annular conservative torus homeomorphisms, and shows this does not extend to eventually annular cases.

Findings

Existence of irrational rotation factors is characterized by bounded deviations.

A well-defined rotation speed in some rational direction is necessary.

Counterexample shows the characterization fails for eventually annular homeomorphisms.

Abstract

We provide an equivalent characterisation for the existence of one-dimensional irrational rotation factors of conservative torus homeomorphisms that are not eventually annular. It states that an area-preserving non-annular torus homeomorphism $f$ is semiconjugate to an irrational rotation $R_\alpha$ of the circle if and only if there exists a well-defined speed of rotation in some rational direction on the torus, and the deviations from the constant rotation in this direction are uniformly bounded. By means of a counterexample, we also demonstrate that a similar characterisation does not hold for eventually annular torus homeomorphisms.