Minimal Equicontinuous Actions on Stone Spaces

TL;DR

This paper investigates minimal equicontinuous actions on Stone spaces, called subodometers and odometers, characterizing their eigenvalues, inverse limit structures, and factor relationships without assuming metrizability or specific group properties.

Contribution

It introduces a comprehensive framework for understanding subodometers and odometers on Stone spaces, including invariants, inverse limit representations, and universal and maximal factors.

Findings

Eigenvalues form a complete invariant for subodometers

Odometers are characterized by intersection stability of eigenvalues

Existence of universal and maximal factors for subodometers and odometers

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.