Group actions on treelike compact spaces
We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler (1990) 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 (for example, dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.
Keywordsamenable group dendrite dendron fragmentability median pretree proximal action Rosenthal Banach space tame dynamical system
MSC(2010)Primary 54H20 secondary 54H15, 22A25
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This work was supported by the Israel Science Foundation (Grant No. ISF 668/13). Thanks are due to Nicolas Monod for enlightening conversations concerning tameness of actions on dendrites. We also thank him and the organizers of the conference entitled “Structure and Dynamics of Polish Groups”, held at the CIB in Lausanne, March 2018, for the invitation to participate in this conference. The successful conference contributed a great deal to the progress of this work. Thanks are due also to Jan van Mill for his helpful advice.
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