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Uniqueness of a topological Furstenberg system

Ioannis Kousek, Vicente Saavedra-Araya

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.

Uniqueness of a topological Furstenberg system

Abstract

Given a semigroup and a bounded function , a topological Furstenberg system of is a topological dynamical system that encodes the dynamical behaviour of . We show that 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.

Paper Structure

This paper contains 14 sections, 24 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Constructions of Furstenberg Systems
  5. Symbolic constructions and other examples
  6. Ultrafilters and the maximal Furstenberg system
  7. Uniqueness of Furstenberg Systems
  8. Uniqueness
  9. Joint Furstenberg systems of multiple functions
  10. Isomorphic sets
  11. Distinguishing the left shift from Furstenberg Systems
  12. Minimal Systems, Minimal Sets, and Topological Recurrence
  13. Minimality in systems and sets
  14. Sets of topological recurrence

Key Result

Theorem 1.2

Let $G$ be any semigroup and $f:G\to {\mathbb C}$ be bounded. Then, any two Furstenberg systems of $f$ are topologically isomorphic. Moreover, any such system is a factor of any generalised Furstenberg system of $f$ and there is a maximal generalised Furstenberg system of $f$ that contains all the o

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 3.3
  • ...and 42 more