Some examples of tame dynamical systems answering questions of Glasner and Megrelishvili

Abstract:

Glasner and Megrelishvili [Trans. Amer. Math. Soc. 375 (2022), pp. 4513–4548] proved that every continuous action of a topological group $G$ on a dendrite $X$ is tame. We produce two examples of an action on a dendrite which is not $\mathrm {tame}_1$, answering a question they raised. We then show that actions on dendrites have $\beta$-rank at most $2$ and produce examples of tame metric dynamical systems of $\beta$-rank $\alpha$ for any $\alpha <\omega _1$, answering another question of Glasner and Megrelishvili.