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Glasner and Megrelishvili proved that every continuous action of atopological group $G$ on a dendrite $X$ is tame. We produce two examples of anaction on a dendrite which is not $\mathrm{tame}_1$, answering a question theyraised. 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 andMegrelishvili.

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