Groups Acting on Dendrons

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.

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“…Except for the extreme proximality claim the following results were proven (independently) in [38], [27], [7] and [39].…”

Section: Some Consequences Of the Tameness Of Actions On Dendritesmentioning

confidence: 88%

Group actions on treelike compact spaces

Glasner

Megrelishvili

2019

Sci. China Math.

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“…We notice that (i) and (ii) are stated in [20] as Corollary 2.15.1, and the proof of (iii), as noted by Malyutin [14], can be obtained as a consequence of Lemma 2.7 combined with Corollary 2.14 of [20].…”

Section: Set Up and Toolsmentioning

confidence: 92%

A note on actions with finite orbits on dendrites

Abdalaoui,

Naghmouchi

2019

Preprint

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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.

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“…Shi proved that every minimal group action on dendrites is strongly proximal and the acting group cannot be amenable (see [14,15]). 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]).…”

Section: Introductionmentioning

confidence: 89%