Topological conjugation classes of tightly transitive subgroups of ${\rm Homeo}_+{(\mathbb R)}$

Enhui Shi¹ ; Lizhen Zhou¹

¹ School of Mathematical Sciences Soochow University Suzhou 215006, P.R. China

Colloquium Mathematicum, Tome 145 (2016) no. 1, pp. 111-120

Résumé

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.

Détail

DOI : 10.4064/cm6627-1-2016

Keywords: mathbb real line homeo mathbb orientation preserving homeomorphism group mathbb subgroup homeo mathbb called tightly transitive there point orbit dense subgroups infty have property paper each integer determine topological conjugation classes tightly transitive subgroups homeo mathbb which isomorphic mathbb have countably many nontransitive points