Induced dynamics and quasifactors for minimal equicontinuous actions on Stone spaces

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.

💡 Research Summary

The paper investigates minimal equicontinuous actions of a discrete group (G) on a Stone space (X). Such actions are called subodometers; those that arise from a genuine group rotation are called odometers. The authors study the dynamics induced on two natural “higher‑level’’ spaces associated with (X): the hyperspace (\mathcal H(X)) of all non‑empty closed subsets equipped with the Vietoris topology, and the space (\mathcal M(X)) of regular Borel probability measures equipped with the weak‑* topology.

Main structural results.

1. Decomposition into subodometers. Because ((X,G)) is equicontinuous, the induced actions ((\mathcal H(X),G)) and ((\mathcal M(X),G)) are also equicontinuous. Consequently each of them splits as a disjoint union of minimal components, and every minimal component is itself a subodometer (the hyperspace remains a Stone space, while the measure space is compact and connected).

2. Characterisation of odometers via induced dynamics. For an infinite subodometer ((X,G)) the following are equivalent:

   • ((\mathcal H(X),G)) decomposes only into factors of ((X,G));
   • ((\mathcal M(X),G)) decomposes only into factors of ((X,G));
   • ((X,G)) is actually an odometer (i.e. a rotation on a compact group).
     Thus the nature of the induced dynamics completely detects whether a subodometer is a genuine rotation. The equivalence fails for finite subodometers, and explicit counter‑examples are discussed.

3. Quasifactors and regular recurrence. The paper introduces several notions of regular recurrence to bridge the gap between the original system and its induced systems. Using these notions, the authors prove that every (\mathcal H)-quasifactor of a subodometer is conjugate to a factor of ((X,G)); moreover, when ((X,G)) is an odometer, every (\mathcal H)-quasifactor coincides with a factor. An analogous theory is developed for (\mathcal M)-quasifactors: any factor of ((X,G)) appears as an (\mathcal M)-quasifactor, and for odometers the two notions are identical.

4. Disjointness criteria. Two minimal actions ((X,G)) and ((Y,G)) are disjoint (the product action is minimal) if and only if every finite factor of one is disjoint from every finite factor of the other. This is reformulated in group‑theoretic language: letting (\operatorname{Eig}(X,G)) and (\operatorname{Eig}(Y,G)) denote the collections of finite‑index subgroups that appear as eigenvalues (i.e. give rise to finite factors), disjointness is equivalent to the condition (\Lambda\Gamma=G) for all (\Lambda\in\Eig(X,G)) and (\Gamma\in\Eig(Y,G)). The paper also shows that having no non‑trivial common factor is equivalent to the same group‑generation condition.

5. Classification of actions disjoint from all subodometers. The authors prove that a minimal action is disjoint from every subodometer (hence from every odometer and from every finite minimal action) precisely when it has no non‑trivial finite factors. Equivalently, its maximal equicontinuous factor (MEF) is a connected space. This result generalises the classical fact that a minimal action admitting an invariant probability measure is weakly mixing iff it is disjoint from all equicontinuous actions.

Methodological contributions.
The work combines tools from several areas:

• Topology of Stone spaces and Vietoris hyperspaces, guaranteeing that (\mathcal H(X)) inherits the zero‑dimensional, totally disconnected structure of (X).
• Measure‑theoretic dynamics, using the weak‑* topology on (\mathcal M(X)) and the unique invariant measure of minimal equicontinuous systems.
• Group theory, especially the lattice of finite‑index subgroups of (G) and the notion of eigenvalues as finite‑index quotients.
• The theory of Ellis semigroups and quasifactors, extending Glasner’s original framework to the setting of subodometers.

By analysing the induced dynamics on (\mathcal H(X)) and (\mathcal M(X)), the paper provides a unified picture: the “higher‑level’’ systems reflect the algebraic nature of the original action, and the presence or absence of non‑trivial finite factors completely determines the interaction with all subodometers. This yields a clear dichotomy between odometers (pure rotations) and more general subodometers, and offers new criteria for disjointness and weak mixing in the broader context of actions on Stone spaces.

Overall, the article advances the understanding of minimal equicontinuous actions by revealing how their hyperspace and measure‑space extensions encode the same dynamical information, and by establishing precise algebraic–topological conditions that characterize odometers, quasifactors, and disjointness phenomena.