Minimal sets of non-resonant torus homeomorphisms

TL;DR

This paper classifies the minimal sets of non-resonant torus homeomorphisms, extending classical circle dynamics results to two-dimensional tori with irrational rotation vectors.

Contribution

It provides a classification of minimal sets for non-resonant torus homeomorphisms, a two-dimensional analogue of classical circle homeomorphism results.

Findings

Classification of minimal sets for non-resonant torus homeomorphisms

Extension of circle dynamics results to two-dimensional tori

Identification of conditions for minimality in torus homeomorphisms

Abstract

As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.