Minimal sets and orbit space for group actions on local dendrites

Abstract

We consider a group $G$ acting on a local dendrite $X$ (in particular on a graph). We give a full characterization of minimal sets of $G$ by showing that any minimal set $M$ of $G$ (whenever $X$ is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. If $X$ is a graph different from a circle, such a minimal $M$ is a finite orbit. These results extend those of the authors for group actions on dendrites. On the other hand, we show that, for any group $G$ acting on a local dendrite $X$ different from a circle, the following properties are equivalent: (1) ($G, X$) is pointwise almost periodic. (2) The orbit closure relation $R = \{(x, y)\in X\times X: y\in \overline{G(x)}\}$ is closed. (3) Every non-endpoint of $X$ is periodic. In addition, if $G$ is countable and $X$ is a local dendrite, then ($G, X$) is pointwise periodic if and only if the orbit space $X/G$ is Hausdorff.