Uniqueness of a topological Furstenberg system

TL;DR

This paper proves the uniqueness of topological Furstenberg systems associated with functions on semigroups, extending measurable case results and characterizing sets with minimal systems.

Contribution

It establishes the topological uniqueness of Furstenberg systems for semigroup functions and characterizes sets with minimal systems and recurrence.

Findings

Furstenberg systems are unique up to topological isomorphism.

Necessary and sufficient conditions for subsets to have isomorphic Furstenberg systems.

Sets with minimal Furstenberg systems are a special class of syndetic sets.

Abstract

Given a semigroup $G$ and a bounded function $f: G \to \mathbb{C}$, a topological Furstenberg system of $f$ is a topological dynamical system $\mathbb{X}=(X, (T_g)_{g \in G})$ that encodes the dynamical behaviour of $f$. We show that $\mathbb{X}$ is unique up to topological isomorphism, thus providing a topological analogue of the measurable case established by Bergelson and Ferr\'e Moragues for amenable semigroups. We also provide necessary and sufficient conditions for subsets of a group to have isomorphic Furstenberg systems. In addition, we study sets with minimal Furstenberg systems and identify them as a special subclass of dynamically syndetic sets. Moreover, we use this notion to obtain a new characterization of sets of topological recurrence.