Minimal Hyperspace Actions of Homeomorphism Groups of H-homogeneous Spaces

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)), Homeo(X)), where Exp(X) stands for the hyperspace comprising the closed subsets of X equipped with the Vietoris topology. The importance of this dynamical system stems from Uspenskij’s characterization of the universal ambit of G = Homeo(X). The results apply to X = C the Cantor set, the generalized Cantor sets X = {0, 1} for non-countable cardinals κ, and to several other spaces. A particular interesting case is X = ω∗ = βω \ ω, where βω denotes the Stone-Čech compactification of the natural numbers. This space, called the corona or the remainder of ω, has been extensively studied in the fields of set theory and topology.