2004/06/09 by Eli Glasner, Michael Megrelishvili, Glasner, Eli +1 · 4 citations
Mathematics · #37B05 #43A60 #46B22 #54H15 #54H20 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/0406192
openalex publication_date 2004/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 by Megrelishvili (2003), 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-Nikodym property. We call such a system a Radon-Nikodym 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. We also show that for symbolic systems the RN property is equivalent to having a countable phase space; and that any Z-dynamical system (f,X), where X is either the unit interval or the unit circle and f: X→ X is a homeomorphism, is an RN system. Using fragmentability and Namioka's theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup of a compact metrizable HNS system is a separable Rosenthal compact, hence of cardinality less than or equal c. We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorucević dichotomy concerning Rosenthal compacts.