Minimal hyperspace actions of homeomorphism groups of h-homogeneous spaces

TL;DR

This paper classifies all minimal sub-systems of a complex dynamical system involving homeomorphism groups of h-homogeneous spaces, using combinatorial and Ramsey-theoretic methods, with implications for universal ambit characterization.

Contribution

It provides a complete list of minimal sub-systems for the hyperspace dynamical system of h-homogeneous spaces, extending to various important spaces like the Cantor set and beta(omega).

Findings

Complete classification of minimal sub-systems.

Application of dual Ramsey theorem and combinatorial analysis.

Results encompass spaces like the Cantor set and beta(omega).

Abstract

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.