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

TL;DR

This paper explores a hierarchy within tame dynamical systems based on the topological complexity of their enveloping semigroups, extending Todorcevic's trichotomy to classify systems by separability properties.

Contribution

It introduces a new hierarchy of tame systems based on the topological properties of their enveloping semigroups, specifically first countability and hereditary separability.

Findings

Defined classes Tame_1 and Tame_2 within tame systems.

Provided examples illustrating properties of these classes.

Analyzed the hierarchy's implications for dynamical systems.

Abstract

Todorcevi\'{c}' 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 $\mathrm{Tame}_\mathbf{2} \subset \mathrm{Tame}_\mathbf{1} \subset \mathrm{Tame},$ 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.