Induced dynamics and quasifactors for minimal equicontinuous actions on Stone spaces

TL;DR

The paper develops a detailed framework for minimal equicontinuous actions on Stone spaces (subodometers) by analyzing their hyperspaces ${\mathcal{H}}(X)$ and spaces of regular Borel probability measures ${\mathcal{M}}(X)$. It establishes that both hyperspace and measure dynamics decompose into disjoint subodometers, and proves a sharp criterion: an infinite subodometer is an odometer if and only if ${\mathcal{H}}(X)$ (respectively ${\mathcal{M}}(X)$) decomposes into factors of $(X,G)$. The authors introduce ${\mathcal{H}}$- and ${\mathcal{M}}$-quasifactors, connect them to settled eigengroups, and show that for odometers these quasifactors coincide with factors; for infinite non-odometer subodometers, finite quasifactors can fail to be factors. They further develop induced dynamics on ${\mathcal{M}}(X)$, provide a parallel analysis to hyperspaces, and give comprehensive disjointness criteria, including a test via scales and eigenvalues, culminating in a characterization of actions disjoint from all subodometers by the connectedness of the maximal equicontinuous factor. Overall, the work links eigenstructure, inverse limit representations, and regular recurrence to precise structural decompositions and disjointness phenomena for subodometers and their induced dynamics.

Abstract

A minimal equicontinuous action of a group $G$ on a Stone space $X$ is called a subodometer. If such a subodometer arises from a group rotation, we refer to it as an odometer. For subodometers $(X,G)$ we show that the hyperspace $\mathcal{H}(X)$ - given by all closed subsets of $X$ and the Vietoris topology - decomposes into subodometers. We show that an infinite subodometer is an odometer if and only if $\mathcal{H}(X)$ decomposes into factors of $(X,G)$. Similarly, we consider $\mathcal{M}(X)$, the space of regular Borel probability measures equipped with the weak-* topology. We show that for a subodometer $(X,G)$ also the connected space $\mathcal{M}(X)$ decomposes into subodometers. We prove that an infinite subodometer $(X,G)$ is an odometer if and only if $\mathcal{M}(X)$ decomposes into factors of $(X,G)$. For this, we study different notions of regular recurrence. Furthermore, we study the disjointness of minimal actions to subodometers and show that this disjointness can be detected from the pairwise disjointness of finite factors. Using this we prove that a minimal action is disjoint from all subodometers if and only if it has a connected maximal equicontinuous factor.

Theorems & Definitions (97)

• Definition 1.1
• Theorem 1.2
• Theorem
• Theorem
• Theorem : \ref{['the:disjointnessCharacterization']}
• Theorem : \ref{['the:characterizationNNCFsubodometers']}
• Theorem : \ref{['the:characterizationDisjointnessToSubodometers']}
• Remark 2.1
• Proposition 2.2
• proof