Hereditarily non-sensitive dynamical systems and linear representations
| dc.creator | Glasner, Eli | |
| dc.creator | Megrelishvili, Michael | |
| dc.date | 2004-06-09 | |
| dc.date | 2006-01-11 | |
| dc.date.accessioned | 2026-07-07T06:36:50Z | |
| dc.date.available | 2026-07-07T06:36:50Z | |
| dc.description | 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\to 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. | |
| dc.description | In the new version there are some improvements and many minor corrections | |
| dc.identifier | https://arxiv.org/abs/math/0406192 | |
| dc.identifier | http://arxiv.org/abs/math/0406192 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100218 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 54H20; 54H15; 37B05; 43A60; 46B22 | |
| dc.title | Hereditarily non-sensitive dynamical systems and linear representations | |
| dc.type | text |