Minimal Equicontinuous Actions on Stone Spaces

TL;DR

The paper analyzes minimal equicontinuous $G$-actions on Stone spaces, introducing subodometers (general equicontinuous actions) and odometers (group rotations). It proves that the eigenvalue set ${\rm Eig}(X,G)$ serves as a complete conjugacy invariant for subodometers and characterizes odometers by a filter-like eigenstructure, establishing inverse-limit representations and factorization results. A comprehensive lattice-theoretic framework for eigensets is developed, showing existence of a universal odometer and maximal subodometer/odometer factors, and linking these to the Ellis semigroup $E(X)$. The work also provides explicit constructions of scales for minimal common extensions and maximal factors, and analyzes metrizability via scales, chain representations, and tile bases, with implications for the structure and classification of minimal equicontinuous dynamics on non-metrizable Stone spaces.

Abstract

In this article we study minimal equicontinuous actions on Stone spaces, which we call \emph{subodometers}, and do neither assume that the space is metrizable, nor any assumptions on the acting group. We show that the set of eigenvalues is a complete invariant for subodometers. Furthermore, we characterize minimal rotations on Stone spaces, which we call \emph{odometers}, via the intersection stability of their sets of eigenvalues. We show that any non-empty family of odometers allows for a minimal common extension and a maximal common factor, that both are odometers and that they are unique up to conjugacy. We provide examples that a similar statement does not hold for subodometers. We show that subodometers are given as inverse limits of minimal finite actions, that odometers are given as inverse limits of minimal finite rotations, and present how the minimal common extension and the maximal common factor of a non-empty family of odometers can be represented as an inverse limit. We establish that a minimal action $X$ is a subodometer if and only if its Ellis semigroup $E(X)$ is an odometer, and present how an inverse limit representation of $E(X)$ can be derived from the representation of $X$. Furthermore, we establish the existence of a universal odometer that has all subodometers as factors; as well as the existence of a maximal subodometer factor, and a maximal odometer factor of a given minimal action.

Theorems & Definitions (140)

• Definition 1.1
• Theorem : Proposition \ref{['pro:factorsOfSubodometers']} and Theorem \ref{['the:eigConjugacyInvariant']}
• Theorem 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Lemma 3.1