Minimal sets and orbit spaces for group actions on local dendrites

Abstract

We consider a group G acting on a local dendrite X (in particular on a graph). We give a full characterization of minimal sets of G by showing that any minimal set M of G (whenever X is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. If X is a graph different from a circle, such a minimal M is a finite orbit. These results extend those of the authors for group actions on dendrites. On the other hand, we show that, for any group G acting on a local dendrite X different from a circle, the following properties are equivalent: (1) (G, X) is pointwise almost periodic.(2) The orbit closure relation R = {(x, y) ∈ X × X : y ∈ G(x)} is closed. (3) Every non-endpoint of X is periodic. In addition, if G is countable and X is a local dendrite, then (G, X) is pointwise periodic if and only if the orbit space X/G is Hausdorff.

Group actions on treelike compact spaces

“…Of course every self homeomorphism T : X → X of a dendrite X is monotone. We can now augment some of Naghamouchi's results in [35] as follows (see also [31] and [32]): Theorem 7.5. Let T : X → X be a self homeomorphism of a dendrite X and consider the Z-system (T, X).…”

A note on actions with finite orbits on dendrites

“…For dendrite, H. Marzougui and the second author classify the minimal sets of the group action on dendrite [17] and local dendrites [18]. This classification is analogue to the case of the circle.…”