Minimal sets and orbit space for group actions on local dendrites

TL;DR

This paper characterizes minimal sets for group actions on local dendrites, showing they are finite or Cantor sets or circles, and explores conditions for pointwise almost periodicity and orbit space properties.

Contribution

It extends previous results by fully characterizing minimal sets and establishing equivalences for pointwise almost periodicity on local dendrites.

Findings

Minimal sets are finite, Cantor, or circle

On non-circular graphs, minimal sets are finite orbits

Pointwise almost periodicity is equivalent to a closed orbit relation

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.