Group action on local dendrites

Abstract

Let G be a group acting by homeomorphisms on a local dendrite X with countable set of endpoints. In this paper, it is shown that any minimal set M of G is either a finite orbit, or a Cantor set or a circle. Furthermore, we prove that if G is a finitely generated group, then the flow (G,X) is a pointwise recurrent flow if and only if one of the following two statements holds:(1)X=S1, and (G,S1) is a minimal flow conjugate to an isometric flow, or to a finite cover of a proximal flow;(2)(G,X) is pointwise periodic.