# A note on actions with finite orbits on dendrites

## TL;DR

This paper investigates group actions on dendrites, showing that actions with finite orbits are equicontinuous on minimal sets, and characterizes invariant measures, extending results to dendrons and including non-amenable groups.

## Contribution

It establishes conditions under which group actions on dendrites are equicontinuous and characterizes invariant measures, extending previous results to broader classes of groups and spaces.

## Key findings

- Actions with finite orbits are equicontinuous on minimal sets.
- Amenable and Thompson groups have equicontinuous actions on dendrites.
- Identifies non-amenable groups with finite orbit actions on dendrites.

## Abstract

It is shown that the restriction of the action of any group with finite orbit on the minimal sets of dendrites is equicontinuous. Consequently, we obtain that the action of any amenable group and Thompson group on dendrite restricted on minimal sets is equicontinuous. We further provide a class of non-amenable groups whose action on dendrites has finite orbit. We extend also some of our results to dendron. We further give a characterization of the set of invariant probability measures and its extreme points.