On metrizable enveloping semigroups

Abstract

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 non-sensitive; (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.

Independence in topological and C*-dynamics

“…Section 7 investigates tame extensions of minimal systems. Under the hypothesis of an Abelian acting group, we establish in Theorem 7.15 a proximal-equicontinuous decomposition that answers a question of Glasner from [33] and show in Theorem 7.19 that tame minimal systems are uniquely ergodic. These theorems generalize results from [41] which cover the metrizable null case.…”

“…The concept of tameness was introduced by Köhler in [54] under the term regularity. Here we are following the terminology of [33]. Actually in [33] the system (X, G) is defined to be tame if its enveloping semigroup is separable and Fréchet, which is equivalent to our geometric definition when X is metrizable.…”

“…The transfer of these ideas to the dynamical realm was initiated by Köhler, who applied the Bourgain-Fremlin-Talagrand dichotomy for spaces of Baire class one functions to obtain a corresponding statement for enveloping semigroups of continuous interval maps [54]. This dynamical Bourgain-Fremlin-Talagrand dichotomy was recently extended by Glasner and Megrelishvili to general metrizable systems [34] (see also [33]). The dichotomy hinges on the notion of tameness, which was introduced by Köhler (under the term regularity) and refers to the absence of orbits of continuous functions which contain an infinite subset equivalent to the standard basis of 1 .…”

Representations of dynamical systems on Banach spaces not containing $l_{1}$

“…Thus, either E(X) (although not necessarily metrizable) has a nice topological structure, or it is as unruly as possible containing a copy of βN. As to the metrizability of E(X), recent results in [13] and [16] assert that E(X) is metrizable iff the metric compact G-space X is hereditarily nonsensitive (HNS) iff X is Asplund representable (see Section 3.3.2). A Banach space V is an Asplund space if the dual of every separable Banach subspace is separable (see Remarks 2.2(4)).…”

New algebras of functions on topological groups arising from G-spaces

“…( [14], a metric dynamical system (G, X) is RN iff its enveloping semigroup is metrizable. Now if S is metrizable then it follows that the transitive system (G, S) is RN and therefore trivial.…”