A Classification of Minimal Sets of Torus Homeomorphisms

TL;DR

This paper classifies minimal sets of torus homeomorphisms based on the structure of their complement, providing a framework for understanding their dynamics and special cases like non-wandering maps and Anosov homotopies.

Contribution

It offers a comprehensive classification of minimal sets on the torus, linking their structure to dynamical properties and homotopy types, with strengthened results for non-wandering maps.

Findings

Minimal sets are classified into three types based on their complement structure.

In non-wandering cases, minimal sets are either periodic or related to circloids or Cantor sets.

Certain homotopy classes exclude some types of minimal sets, especially for Anosov and identity homotopies.

Abstract

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.