On metrizable enveloping semigroups

Eli Glasner, Michael Megrelishvili, Vladimir V. Uspenskij

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g G, in the compact space X X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d G (x, y) := sup{d(gx, gy): g G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.

Original languageEnglish
Pages (from-to)317-332
Number of pages16
JournalIsrael Journal of Mathematics
Volume164
DOIs
StatePublished - Mar 2008

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