[Paper Review] Linear representations of hereditarily non-sensitive dynamical systems

Eli Glasner, Michael Megrelishvili|arXiv (Cornell University)|Jun 9, 2004

Advanced Operator Algebra Research23 references29 citations

TL;DR

This paper introduces Radon-Nikodým (RN) systems—compact G-dynamical systems linearly representable in dual Banach spaces with the Radon-Nikodým property—and establishes that for metrizable systems, RN, hereditarily non-sensitive (HNS), and hereditarily almost equicontinuous (HAE) classes coincide. The key contribution is a characterization of HNS systems via enveloping semigroups as separable Rosenthal compacts of size at most continuum, and a proof that discrete countable groups admit at most two-to-one metric factors in such systems.

ABSTRACT

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 [44] 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-Nikod´ym property. We call such a system a Radon-Nikod´ym 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 and show that the enveloping semigroup of a compact metrizable HNS system is a separable Rosenthal compact, hence of cardinality ≤ 2 ℵ0. Moreover, applying a theorem of Todor˘cević we show, for discrete countable acting groups, that it admits an at most two-to-one metric factor.

Motivation & Objective

• To characterize compact G-dynamical systems that admit linear representations in dual Banach spaces with the Radon-Nikodým property.
• To investigate the relationship between RN systems and previously studied classes such as WAP and LE (locally equicontinuous) systems.
• To determine whether Glasner-Weiss examples of recurrent-transitive, locally equicontinuous but not WAP cascades are RN systems.
• To clarify the conditions under which symbolic systems are RN, particularly in relation to countable phase spaces.
• To provide a topological and functional-analytic characterization of HNS systems using enveloping semigroups and Namioka’s theorem.

Proposed method

• Utilize the weak∗-compact embedding of any compact G-system into the dual of a Banach space V ∗, with V chosen to have the Radon-Nikodým property for RN systems.
• Apply fragmentability and Namioka’s theorem to analyze the structure of enveloping semigroups of HNS systems.
• Use Todorčević’s theorem to derive metric factorization results for discrete countable acting groups.
• Establish equivalence between RN and HNS/HAE classes in metrizable compact G-systems via topological and functional-analytic techniques.
• Analyze symbolic systems by linking the RN property to the cardinality of the phase space, showing equivalence when the space is countable.
• Employ the theory of Rosenthal compacta to bound the size of the enveloping semigroup of HNS systems, proving it is at most 2ℵ₀.

Experimental results

Research questions

• RQ1 Do all hereditarily non-sensitive (HNS) systems admit a linear representation in a dual Banach space with the Radon-Nikodým property, i.e., are they RN systems?
• RQ2 Is there a complete classification of RN systems in terms of topological and dynamical properties such as equicontinuity and sensitivity?
• RQ3 Are the Glasner-Weiss examples of recurrent-transitive, locally equicontinuous but not weakly almost periodic cascades examples of RN systems?
• RQ4 For symbolic systems, is the RN property equivalent to having a countable phase space?
• RQ5 What is the structure of the enveloping semigroup of a metrizable HNS system, and can it be bounded in size?

Key findings

• For metrizable compact G-systems, the classes of RN, HNS, and HAE systems are equivalent.
• The enveloping semigroup of any compact metrizable HNS system is a separable Rosenthal compact, hence of cardinality at most 2ℵ₀.
• The Glasner-Weiss examples of recurrent-transitive, locally equicontinuous but not WAP cascades are shown to be RN systems.
• For symbolic systems, the RN property holds if and only if the phase space is countable.
• Any Z-dynamical system (f, X) with X being the unit interval or the unit circle and f a homeomorphism is an RN system.
• For discrete countable acting groups, every HNS system admits an at most two-to-one metric factor, as established via Todorčević’s theorem.