Topological conjugation classes of tightly transitive subgroups of ${\rm Homeo}_+{(\mathbb R)}$
Volume 145 / 2016
Colloquium Mathematicum 145 (2016), 111-120
MSC: Primary 37B05; Secondary 57S25.
DOI: 10.4064/cm6627-1-2016
Published online: 22 April 2016
Abstract
Let $\mathbb R$ be the real line and let ${\rm Homeo}_+(\mathbb R)$ be the orientation preserving homeomorphism group of $\mathbb R$. Then a subgroup $G$ of ${\rm Homeo}_+(\mathbb R)$ is called tightly transitive if there is some point $x\in X$ such that the orbit $Gx$ is dense in $X$ and no subgroups $H$ of $G$ with $|G:H|=\infty $ have this property. In this paper, for each integer $n \gt 1$, we determine all the topological conjugation classes of tightly transitive subgroups $G$ of ${\rm Homeo}_+(\mathbb R)$ which are isomorphic to $\mathbb Z^n$ and have countably many nontransitive points.
