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.