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

Abstract

Glasner and Megrelishvili 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.