Equicontinuity of minimal sets for amenable group actions on dendrites

TL;DR

This paper proves that for amenable groups acting on dendrites, minimal sets are either finite or Cantor sets, and the group action on these sets is equicontinuous, revealing a clear structure of such dynamical systems.

Contribution

It establishes the equicontinuity of group actions on minimal sets in dendrites and classifies these sets as finite or Cantor sets, advancing understanding of group dynamics on dendritic spaces.

Findings

Minimal sets are either finite or homeomorphic to the Cantor set.

Group actions on minimal sets are equicontinuous.

The structure of minimal sets under amenable group actions is characterized.

Abstract

In this note, we show that if $G$ is an amenable group acting on a dendrite $X$, then the restriction of $G$ to any minimal set $K$ is equicontinuous, and $K$ is either finite or homeomorphic to the Cantor set.