The universal minimal space of the homeomorphism group of a \(h\)-homogeneous space. (English) Zbl 1279.37008

Glasner, Eli; Gutman, Yonatan

Bowen, Lewis (ed.) et al., Dynamical systems and group actions. Dedicated to Anatoli Stepin on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-6922-2/pbk; 978-0-8218-8539-0/ebook). Contemporary Mathematics 567, 105-117 (2012).

A compact zero-dimensional space is called \(h\)-homogeneous if all nonempty clopen subsets are homeomorphic. Equivalently, \(X\) is the Stone space of a homogeneous Boolean algebra. Most topologically homogeneous zero-dimensional compact spaces are \(h\)-homogeneous, but not all. In this interesting paper, the authors show that the universal minimal flow of the homeomorphism group of an \(h\)-homogeneous compact space is the compact space of maximal chains of nonempty closed subsets of \(X\). This space was introduced by Uspenskij. Several known results are consequences of the main result in this paper.
For the entire collection see [Zbl 1237.37004].

Reviewer: Jan van Mill (Amsterdam)

MSC:

37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
06E15	Stone spaces (Boolean spaces) and related structures
54H10	Topological representations of algebraic systems
54H20	Topological dynamics (MSC2010)

Keywords:

universal minimal space; \(h\)-homogeneous; homogeneous Boolean algebra; maximal chains; Cantor sets; dual Ramsey theorem; corona; remainder; Parovicenko space; collapsing algebra