MINIMAL SETS FOR GROUP ACTIONS ON DENDRITES

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 M of all minimal sets in the hyperspace 2(X) (endowed with the Hausdorff metric). We prove that M is closed in 2(X) and that the family F of all finite orbits (when it is non-empty) is dense in M. As a consequence, the union of all minimal sets of G is closed in X.