Induced dynamics and quasifactors for minimal equicontinuous actions on Stone spaces

TL;DR

This paper investigates the structure of minimal equicontinuous actions called subodometers on Stone spaces, analyzing their decompositions into factors via hyperspaces and measure spaces, and characterizing their disjointness properties.

Contribution

It introduces a detailed decomposition framework for subodometers using hyperspaces and measure spaces, and characterizes odometers among subodometers through these decompositions.

Findings

Infinite subodometers are odometers iff their hyperspace decomposes into factors.

Infinite subodometers are odometers iff their measure space decomposes into factors.

Disjointness from all subodometers is characterized by having a connected maximal equicontinuous factor.

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.