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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.)
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