Hereditarily non-sensitive dynamical systems and linear representations

E. Glasner¹ ; M. Megrelishvili²

¹ Department of Mathematics Tel-Aviv University Ramat Aviv, Israel} \emailaut1{[e-mail].i
² Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel

Colloquium Mathematicum, Tome 104 (2006) no. 2, pp. 223-283.

Résumé

For an arbitrary topological group $G$ any compact $G$-dynamical system $(G,X)$ can be linearly $G$-represented as a weak$^*$-compact subset of a dual Banach space~$V^*$. As~was shown in {M1} the Banach space $V$ can be chosen to be reflexive iff the metric system $(G,X)$ is weakly almost periodic (WAP). In the present paper we study the wider class of compact $G$-systems which can be linearly represented as a weak$^*$-compact subset of a dual Banach space with the Radon–Nikodým property. We call such a system a Radon–Nikodým (RN) system. One of our main results is to show that for metrizable compact $G$-systems the three classes: RN, HNS (hereditarily non-sensitive) and HAE (hereditarily almost equicontinuous) coincide. We investigate these classes and their relation to previously studied classes of $G$-systems such as WAP and LE (locally equicontinuous). We show that the Glasner–Weiss examples of recurrent-transitive locally equicontinuous but not weakly almost periodic cascades are actually RN. Using fragmentability and Namioka's theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup $E(X)$ of a compact metrizable HNS $G$-system is a separable Rosenthal compact, hence of cardinality $\le 2^{\aleph_0}$. We investigate a dynamical version of the Bourgain–Fremlin–Talagrand dichotomy and a dynamical version of the Todor\v{c}ević dichotomy concerning Rosenthal compacts.

Détail

DOI : 10.4064/cm104-2-5

Keywords: arbitrary topological group compact g dynamical system linearly g represented weak * compact subset dual banach space * shown banach space chosen reflexive metric system weakly almost periodic wap present paper study wider class compact g systems which linearly represented weak * compact subset dual banach space radon nikod property call system radon nikod system main results metrizable compact g systems three classes hns hereditarily non sensitive hae hereditarily almost equicontinuous coincide investigate these classes their relation previously studied classes g systems wap locally equicontinuous glasner weiss examples recurrent transitive locally equicontinuous weakly almost periodic cascades actually using fragmentability namiokas theorem enveloping semigroup characterization hns systems enveloping semigroup compact metrizable hns g system separable rosenthal compact hence cardinality aleph investigate dynamical version bourgain fremlin talagrand dichotomy dynamical version todor evi dichotomy concerning rosenthal compacts