Uniqueness of a topological Furstenberg system

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 Ferré 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.

26 pages. Comments welcome!