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}^kappa for non-countable cardinals kappa, and to several other spaces. A particular interesting case is X = beta(omega) \ omega, where beta(omega) denotes the Stone-Cech compactification of the natural numbers. This space, called the corona or the remainder of omega, has been extensively studied in the fields of set theory and topology. 
Eli Glasner and Yonatan Gutman
math.DS