Hereditarily non-sensitive dynamical systems and linear representations

• Autorzy: E. Glasner , M. Megrelishvili
• Języki publikacji: EN

Abstrakty

EN

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 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 $ ≤ 2^{ℵ₀}$. We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of the Todorčević dichotomy concerning Rosenthal compacts.

Kategorie tematyczne

• 46B22: Radon-Nikod{\'y}m, Kre{\u\i}n-Milman and related properties
• 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
• 54H20: Topological dynamics
• 54H15: Transformation groups and semigroups
• 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2006
• Tom: 104
• Numer: 2
• Strony: 223-283

Daty

wydano

2006

Twórcy

autor

E. Glasner

• Department of Mathematics, Tel-Aviv University, Ramat Aviv, Israel

autor

M. Megrelishvili

• Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel

Identyfikatory

DOI

10.4064/cm104-2-5