An Extension to the Non-metric Case of a Theorem of Glasner

In [1] the Furstenberg Structure Theorem for flows was extended from the metric to the non-metric case by means of a special construction of minimal flows. We are able to use this construction to generalize another theorem from the metric to the non-metric case: Glasner proved in [4] that if the space of regular Borel probability measures of a flow is distal, then it is equicontinuous, provided the flow is a metric minimal flow. We are able here to remove the metric condition. Let X be a compact Hausdorff space, and let M(X) be the set of regular Borel probability measures on X. M(X) will always be assumed to have the weak-* topology induced as a subset of the dual of C(X), that is, μi → μ ⇐⇒ ∫ f dμi → ∫ f dμ ∀ f ∈ C(X). With this topology, M(X) is compact Hausdorff. Moreover, if X is metric, then so is M(X). A Dirac measure is a measure of the form δx, where δx is defined to be ∫ f dδx = f(x). The function δ : X 7→ M(X) that sends x to δx is a homeomorphism onto its image. We’ll sometimes identify X with δ(X). If π : X 7→ Y is continuous, we define π̂ : M(X) 7→ M(Y ) by ∫ f dπ̂(μ) = ∫ (f ◦ π)dμ. Assume now that (X,T ) is a flow. The action of T on X induces an action of T on M(X) in the following way: first, if f is a measurable function, define tf to be tf(x) = f(xt). Then, define μt as the measure given by ∫ f d(μt) = ∫ (tf) dμ. This is an action and (M(X), T ) is a flow. Definition 1. A (not-necessarily minimal) flow (X,T ) is called strongly distal (or sd for short) if (M(X), T ) is distal. Remark 1. Strongly distal implies distal since X is a closed T -invariant subset of M(X) (by means of the identification X = δX = {δx : x ∈ X})). Lemma 1. If π : X 7→ Y is an epimorphism of T -flows and (X,T ) is strongly distal, then so is (Y, T ). Proof. If π : X → Y is a homomorphism onto, then so is π̂ : M(X) 7→ M(Y ). Thus, M(X) distal implies that M(Y ) is distal. Definition 2. Let S := {H : H is a countable subgroup of T}. Let ρ be a Work supported in part by grants from CONICET, CONICOR and SECYT. The author wishes to thank Professor Robert Ellis for his constant encouragement and support.