Group action with finite orbits on local dendrites

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