Hereditarily non-sensitive dynamical systems and linear representations

By Eli Glasner, Michael Megrelishvili. 2004

88 citations recorded by the source corpus.

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 ≤ 2א0 . We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorcevic dichotomy concerning Rosenthal compacts.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5a518f27a518af19f37ae145203f32f542718de

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A note on sensitivity of semigroup actions (2006, background): “The present form is a generalized version of existing definitions for cascades (see also [8, 7]).”
Local entropy theory (2009, background): “In [37], hereditarily non-sensitive systems (HNSs) were shown to be tame and in [38] Glasner et al showed that metric HNSs are exactly those metric systems whose enveloping semigroup is metrizable.”
Representations of dynamical systems on Banach spaces not containing $l_1$ (2008, background): “all continuous real valued bounded functions on a topological space Xwill be denoted by C(X). The material in this section is mostly well known. For more details and undeﬁned concepts see for example [13, 12, 14]. 3.1. Cyclic G-spaces. A function f∈ C(X) on a G-space Xcomes from a compact G-space Y ”
Independence in topological and C*-dynamics (2006, unknown): “1”