Todorc̆ević’ trichotomy and a hierarchy in the class of tame dynamical systems

Glasner, Eli; Megrelishvili, Michael

doi:10.1090/tran/8522

Todorc̆ević’ trichotomy and a hierarchy in the class of tame dynamical systems
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Trans. Amer. Math. Soc. 375 (2022), 4513-4548 Request permission

Abstract:

Todorc̆ević’ trichotomy in the class of separable Rosenthal compacta induces a hierarchy in the class of tame (compact, metrizable) dynamical systems $(X,T)$ according to the topological properties of their enveloping semigroups $E(X)$. More precisely, we define the classes \begin{equation*} \mathrm {Tame}_\mathbf {2} \subset \mathrm {Tame}_\mathbf {1} \subset \mathrm {Tame}, \end{equation*} where $\mathrm {Tame}_\mathbf {1}$ is the proper subclass of tame systems with first countable $E(X)$, and $\mathrm {Tame}_\mathbf {2}$ is its proper subclass consisting of systems with hereditarily separable $E(X)$. We study some general properties of these classes and exhibit many examples to illustrate these properties.

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