The realization and classification of topologically transitive group actions on $1$-manifolds

TL;DR

This paper reviews key classification theorems for circle homeomorphisms and introduces a new classification for topologically transitive actions of  groups on the circle, also identifying groups with or without such actions on the line.

Contribution

It presents a novel classification theorem for topologically transitive  group actions on the circle, expanding understanding beyond existing minimal action classifications.

Findings

Classification of minimal circle actions by Ghys

New classification theorem for  group actions on the circle

Identification of groups with/without transitive actions on the line

Abstract

In this report, we first recall the Poincar\'e's classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys' classification theorem for minimal orientation-preserving group actions on the circle. Then we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $\mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.