  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.
