{
  "id": "1204.0432",
  "version": "v5",
  "published": "2012-04-02T15:18:07.000Z",
  "updated": "2013-11-02T18:08:44.000Z",
  "title": "Banach representations and affine compactifications of dynamical systems",
  "authors": [
    "Eli Glasner",
    "Michael Megrelishvili"
  ],
  "comment": "45 pages; Fields institute proceedings dedicated to the 2010 thematic program on asymptotic geometric analysis, M. Ludwig, V.D. Milman, V. Pestov, N. Tomczak-Jaegermann (Editors), Springer, New-York, 2013",
  "categories": [
    "math.DS",
    "math.FA",
    "math.GN"
  ],
  "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 non-sensitive) 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.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-11-02T18:08:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dynamical system",
      "affine compactification",
      "banach representations",
      "enveloping semigroup",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.0432G"
    }
  }
}