## Hereditarily non-sensitive dynamical systems and linear representations

### Volume 104 / 2006

#### Abstract

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.