Hereditarily non-sensitive dynamical systems and linear representations

E. Glasner, and M. Megrelishvili. "Hereditarily non-sensitive dynamical systems and linear representations." Colloquium Mathematicae 104.2 (2006): 223-283. <http://eudml.org/doc/284049>.

@article{E2006,
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 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.},
author = {E. Glasner, M. Megrelishvili},
journal = {Colloquium Mathematicae},
keywords = {G-systems; Ellis semigroups; sensitive dependence; weakly almost periodic},
language = {eng},
number = {2},
pages = {223-283},
title = {Hereditarily non-sensitive dynamical systems and linear representations},
url = {http://eudml.org/doc/284049},
volume = {104},
year = {2006},
}

TY - JOUR
AU - E. Glasner
AU - M. Megrelishvili
TI - Hereditarily non-sensitive dynamical systems and linear representations
JO - Colloquium Mathematicae
PY - 2006
VL - 104
IS - 2
SP - 223
EP - 283
AB - 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.
LA - eng
KW - G-systems; Ellis semigroups; sensitive dependence; weakly almost periodic
UR - http://eudml.org/doc/284049
ER -