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 [45] the Banach space V can be chosen to be reflexive iff the metric system (G, X) is weakly almost periodic (WAP). In this 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 system (RN). One of our main results is to show that for metrizable compact G-systems the three classes: RN, HNS (hereditarily not 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 ≤ 2 ℵ 0 . We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorcević dichotomy concerning Rosenthal compacts.

Abstract. For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of l 1 (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.

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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.

Abstract. It is well known that for a transitive dynamical system (X, f ) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey [2], Silverman [8] and Glasner and Weiss [6]. In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X, f ) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner [4].Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.