Minimal hyperspace actions of homeomorphism groups of h-homogeneous spaces

Abstract

Suppose that X is an 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) denotes 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 the Cantor set, the generalized Cantor sets X = {0,1}^κ for noncountable 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.