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Minimal hyperspace actions of homeomorphism groups of h-homogeneous spaces. (English) Zbl 1273.54036
For $$G$$ a Hausdorff topological group, V. Uspenskij [Topol. Proc. 33, 1–12 (2009; Zbl 1171.54030)] defined a family $$\{X_{\alpha}\mid\alpha\in A\}$$ of compact $$G$$-spaces to be representative if the family of natural maps from the universal ambit of $$G$$ to the enveloping semigroup of $$X_{\alpha}$$ separates points in the universal ambit (a universal ambit is a pointed $$G$$-space that maps homomorphically to any pointed $$G$$-space $$(X,x,G)$$ with a dense orbit of the distinguished point $$x$$). Denote by $$Exp(X)$$ the collection of all non-empty closed subsets of $$X$$ with the Vietoris topology, which is a compact Hausdorff $$G={Homeo}(X)$$-space. Uspenskij showed that for any subgroup $$H< {Homeo}(X)$$ the sequence $$\left(Exp\left(\left(Exp(X)\right)^n\right)\right)_{n\geq1}$$ of $$H$$-spaces (the sequence of hyperspace actions) is representative. For this and other reasons, there is much interest in hyperspace actions. Under the assumption that $$X$$ is a Stone dual of a homogeneous Banach algebra, a complete analysis of the minimal sub-systems of the compact dynamical system $$(Exp(Exp(X)),Homeo(X))$$. These results apply in particular to the Cantor set, to generalized Cantor sets $$\{0,1\}^{\kappa}$$ for uncountable cardinals $$\kappa$$, and to the corona space $$\beta\omega\setminus\omega$$.
##### MSC:
 54H20 Topological dynamics (MSC2010) 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
##### Keywords:
Hyperspace; compact dynamical system; corona
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