Todorcević' trichotomy and a hierarchy in the class of tame dynamical systems

Abstract

Todorcević' 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 ${\mathrm{Tame}}_2\subset {\mathrm{Tame}}_1\subset \mathrm{Tame},$ where ${\mathrm{Tame}}_1$ is the proper subclass of tame systems with first countable $E(X)$, and ${\mathrm{Tame}}_2$ 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.

arXiv comment37 pages, to appear in Trans. AMS