Minimal Equicontinuous Actions on Stone Spaces

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.

💡 Research Summary

This paper develops a comprehensive theory of minimal equicontinuous actions on Stone spaces without assuming metrizability of the phase space or any finiteness conditions on the acting group. The authors introduce the terminology “subodometer” for a minimal equicontinuous action on a Stone space and “odometer” for a minimal rotation (i.e., a group rotation) on such a space. By focusing on the set of eigenvalues – defined as the collection of finite‑index subgroups Γ of the acting group G for which there exists a G‑equivariant factor map onto the finite quotient G/Γ – they obtain a complete conjugacy invariant for subodometers.

The main results can be summarized as follows. First, any factor of a subodometer is again a subodometer, and a subodometer Y is a factor of X if and only if Eig(Y,G)⊆Eig(X,G). Consequently, two subodometers are conjugate precisely when they have identical eigenvalue sets. This establishes Eig(·) as a full invariant for the category of subodometers. Second, a subodometer is metrizable exactly when its eigenvalue set is countable; thus non‑metrizable subodometers arise precisely from groups possessing uncountably many finite‑index subgroups.

For odometers, the authors prove a sharper structural condition: a subodometer (X,G) is an odometer if and only if its eigenvalue set is closed under finite intersections. This mirrors the classical fact that finite‑index normal subgroups form a lattice closed under intersections, and it characterizes minimal rotations among all minimal equicontinuous actions.

The paper then shows that subodometers can be represented as inverse limits of minimal finite actions (nets in the non‑metrizable case, sequences when metrizable). Analogously, odometers are inverse limits of minimal finite rotations, i.e., inverse limits of profinite groups equipped with the natural left G‑action. Using these representations, the authors construct, for any non‑empty family 𝓧 of odometers, a minimal common extension W𝓧 (the supremum) and a maximal common factor V𝓧 (the infimum). Both W𝓧 and V𝓧 are themselves odometers, and they are unique up to conjugacy. Moreover, the lattice of odometers satisfies the modular law: for odometers X₁≤X₂, (X₁∨X)∧X₂ = X₁∨(X∧X₂). For countable families of metrizable odometers the minimal common extension remains metrizable; the maximal common factor of any family of metrizable odometers is always metrizable.

A striking connection with the Ellis semigroup is established: a minimal action (X,G) is a subodometer if and only if its Ellis semigroup E(X) is an odometer. Consequently, the “enveloping odometer” – defined as the infimum of all odometer extensions of (X,G) – coincides (up to conjugacy) with E(X). The authors show how an inverse‑limit description of E(X) can be derived directly from an inverse‑limit description of X. This yields, in particular, that a subodometer is metrizable exactly when its Ellis semigroup is metrizable.

Finally, the paper proves the existence of a universal odometer that factors onto every subodometer, and for any minimal action (X,G) it constructs a maximal subodometer factor and a maximal odometer factor, both uniquely determined up to conjugacy. These constructions are again obtained via inverse‑limit techniques.

Overall, the work provides a unified algebraic–topological framework for classifying minimal equicontinuous actions on arbitrary Stone spaces. By reducing the classification problem to the lattice of finite‑index subgroups of the acting group, the authors extend classical results (which required metrizability or countable groups) to a far broader setting, encompassing non‑metrizable spaces and groups with uncountably many finite quotients. The paper’s blend of uniform space theory, inverse‑limit constructions, and Ellis semigroup analysis offers powerful tools for future investigations in topological dynamics and profinite group actions.