Group actions on treelike compact spaces

Abstract

We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler [37], implies that every action of a topological group G on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of G on a local dendron is null. We then use a more direct method to show that every continuous group action of G on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Helly's selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.

Equicontinuity of minimal sets for amenable group actions on dendrites

“…We obtained the following theorem in this paper, which extends the corresponding result for Z-actions in [11] and answered a question proposed by E. Glasner and M. Megrelishvili in [5]. Theorem 1.1.…”

“…Based on the results obtained by Marzougui and Naghmouchi in [8], Shi and Ye showed that every amenable group action on dendrites always has a minimal set consisting of 1 or 2 points (see [16]), which is also implied by the work of Malyutin and Duchesne-Monod (see [6,3]). Glasner and Megrelishvili showed the extreme proximality of minimal subsystems provided that the group actions on dendrites have no finite orbits (see [5]). For Z actions on dendrites, Naghmouchi proved that every minimal set is either finite or an adding machine (see [11]).…”

Homeomorphism of Hereditarily Locally Connected Continua

“…For instance, Seilder [18] proved that those systems have zero topological entropy. Recently Glasner and Megrelishvili [6] proved that they are even null (i.e their sequence topological entropy is zero). In [14,15] Naghmouchi showed that any ω-limit set is minimal and he showed that the absence of periodic point allows the presence for one and only one minimal set.…”

An alternative for minimal group actions on totally regular curves

“…Recently, some people are interested in studying the alternative phenomena for group actions on curves (continua of one dimension). For example, it is implied by several authors' work that every subgroup of a dendrite homeomorphism group either has a finite orbit or contains a free nonabelian group (see [7,17,11]). One may refer to [12,16,26,27,28] for some related investigations in this direction.…”