Minimal extensions of flows with amenable acting groups
One of the most natural dynamical questions that may be asked of a locally compact group \(\Gamma\) is to determine if it can act minimally on a given space \(X\). Constraining \(\Gamma\) and \(X\) can render this problem tractable: for example, if \(\Gamma=\mathbb{Z}\) and \(X\) is a compact \(2\)-manifold then a minimal action is only possible if \(X\) consists of a finite union of \(2\)-tori and Klein bottles and if \(\Gamma=\mathbb{R}\) then \(X\) can only be a torus. \textit{O. Hajek} [Lect. Notes Math. 318, 140--142 (1973; Zbl 0253.54040)], in discussing a conjecture of Gottschalk, conjectured that there is no minimal continuous flow on a simply connected compact manifold. \textit{H. Chu} and \textit{M. A. Geraghty} [Bull. Am. Math. Soc. 69, 377--381 (1963; Zbl 0112.37901)] related this type of question to the algebraic topology of \(X\), asking in particular if a compact space admitting a minimal continuous flow must have a non-trivial first Čech cohomology group over the integers; a later construction due to \textit{A. Fathi} and \textit{M. R. Herman} [Astérisque 49, 37--59 (1978; Zbl 0374.58010)] shows that in principle we cannot expect any connection between the algebraic topology of \(X\) and the existence of a minimal flow. Here ideas of \textit{S. Glasner} and \textit{B. Weiss} [Isr. J. Math. 34, 321--336 (1979; Zbl 0434.54032)] are used to construct minimal extensions of flows for \(\Gamma\) amenable as fibre-preserving flows on fibre bundles and affine flows on group bundles. These constructions are used to exhibit a large family of groups for which the class of compact second countable spaces that carry minimal actions is closed under countable products.
scientific article; zbMATH DE number 6486893
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