Banach Representations and Affine Compactifications of Dynamical Systems

Eli Glasner, Michael Megrelishvili

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

To every Banach space V we associate a compact right topological affine semigroup ℰ(V). We show that a separable Banach space V is Asplund if and only if ℰ(V) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l 1) if and only if ℰ(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in ℰ(V). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily nonsensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

Original languageEnglish
Title of host publicationAsymptotic Geometric Analysis
Subtitle of host publicationProceedings of the Fall 2010 Fields Institute Thematic Program
EditorsMonika Ludwig, Vladimir Pestov, Vitali Milman, Nicole Tomczak-Jaegermann
Pages75-144
Number of pages70
DOIs
StatePublished - 2013

Publication series

NameFields Institute Communications
Volume68
ISSN (Print)1069-5265

Keywords

  • Affine compactification
  • Affine flow
  • Asplund space
  • Enveloping semigroup
  • Nonsensitivity
  • Right topological semigroup
  • Semigroup compactification
  • Tame system
  • Weakly almost periodic

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