Monotonicity of maximal equicontinuous factors and an application to toral flows

TL;DR

This paper proves that for group actions on locally connected spaces, the maximal equicontinuous factor map is always monotone, and applies this to classify minimal maximal continuous factors of toral homeomorphisms.

Contribution

It establishes the monotonicity of maximal equicontinuous factor maps for group actions on locally connected spaces and classifies minimal factors of toral homeomorphisms.

Findings

Maximal equicontinuous factor maps are monotone for group actions on locally connected spaces.

Minimal maximal continuous factors of toral homeomorphisms are either irrational translations, irrational circle rotations, or trivial.

Provides a classification of minimal factors in the context of toral flows.

Abstract

We show that for group actions on locally connected spaces the maximal equicontinuous factor map is always monotone, that is, the preimages of single points are connected. As an application, we obtain that if the maximal continuous factor of a homeomorphism of the two-torus is minimal, then it is either (i) an irrational translation of the two-torus, (ii) an irrational rotation on the circle or (iii) the identity on a singleton.