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Todorcevi\'{c}' trichotomy in the class of separable Rosenthal compactainduces a hierarchy in the class of tame (compact, metrizable) dynamicalsystems $(X,T)$ according to the topological properties of their envelopingsemigroups $E(X)$. More precisely, we define the classes$\mathrm{Tame}_\mathbf{2} \subset \mathrm{Tame}_\mathbf{1} \subset\mathrm{Tame},$ where $\mathrm{Tame}_\mathbf{1}$ is the proper subclass of tamesystems with first countable $E(X)$, and $\mathrm{Tame}_\mathbf{2}$ is itsproper subclass consisting of systems with hereditarily separable $E(X)$. Westudy some general properties of these classes and exhibit many examples toillustrate these properties.

Todorcevi\'{c}' trichotomy and a hierarchy in the class of tame dynamical systems