Topological conjugation classes of tightly transitive subgroups of $\text{Homeo}_{+}(\mathbb{S}^1)$

TL;DR

This paper classifies topological conjugation classes of certain highly transitive subgroups of circle homeomorphisms that are isomorphic to ^n, providing a comprehensive understanding of their structure.

Contribution

It determines all conjugation classes of tightly transitive, almost minimal subgroups of ext{Homeo}_{+}( ext{S}^1) that are isomorphic to ^n for any n.

Findings

Complete classification of conjugation classes for these subgroups.

Identification of conditions for tight transitivity and almost minimality.

Extension of known results to higher-dimensional free abelian groups.

Abstract

Let $\text{Homeo}_{+}(\mathbb{S}^1)$ denote the group of orientation preserving homeomorphisms of the circle $\mathbb{S}^1$. A subgroup $G$ of $\text{Homeo}_{+}(\mathbb{S}^1)$ is tightly transitive if it is topologically transitive and no subgroup $H$ of $G$ with $[G: H]=\infty$ has this property; is almost minimal if it has at most countably many nontransitive points. In the paper, we determine all the topological conjugation classes of tightly transitive and almost minimal subgroups of $\text{Homeo}_{+}(\mathbb{S}^1)$ which are isomorphic to $\mathbb{Z}^n$ for any integer $n\geq 2$.