Minimal sets for group actions on dendrites

Abstract:

Let $G$ be a group acting by homeomorphisms on a dendrite $X$. First, we show that any minimal set $M$ of $G$ is either a finite orbit or a Cantor set (resp. a finite orbit) when the set of endpoints of $X$ is closed (resp. countable). Furthermore, we prove, regardless of the type of the dendrite $X$, that if the action of $G$ on $X$ has at least two minimal sets, then necessarily it has a finite orbit (and even an orbit consisting of one or two points). Second, we explore the topological and geometrical properties of infinite minimal sets when the action of $G$ has a finite orbit. We show in this case that any infinite minimal set $M$ is a Cantor set which is the set of endpoints of its convex hull $[M]$ and there is no other infinite minimal set in $[M]$. On the other hand, we consider the family $\mathcal {M}$ of all minimal sets in the hyperspace $2^{X}$ (endowed with the Hausdorff metric). We prove that $\mathcal {M}$ is closed in $2^{X}$ and that the family $\mathcal {F}$ of all finite orbits (when it is non-empty) is dense in $\mathcal {M}$. As a consequence, the union of all minimal sets of $G$ is closed in $X$.