Linearisation of conservative toral homeomorphisms

TL;DR

This paper characterizes when conservative toral homeomorphisms are semi-conjugate to irrational rotations, extending classical circle dynamics results to higher dimensions and providing a classification for certain toral homeomorphisms.

Contribution

It offers an equivalent condition for semi-conjugacy to irrational rotations and generalizes Poincare's classification to higher-dimensional tori with specific dynamical properties.

Findings

Characterization of semi-conjugacy to irrational rotations

Extension of Poincare's classification to higher dimensions

Basic classification of non-wandering toral homeomorphisms

Abstract

We give an equivalent characterisation for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincare's classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of non-wandering toral homeomorphisms homotopic to the identitiy.