{
  "id": "1309.6283",
  "version": "v1",
  "published": "2013-09-24T18:33:40.000Z",
  "updated": "2013-09-24T18:33:40.000Z",
  "title": "Ergodic Properties of Discrete Dynamical Systems and Enveloping Semigroups",
  "authors": [
    "A. V. Romanov"
  ],
  "comment": "24 pages",
  "doi": "10.1017/etds.2014.62",
  "categories": [
    "math.DS"
  ],
  "abstract": "For a continuous semicascade on a metrizable compact set $\\Omega $, we consider the weak$^{*}$ convergence of generalized operator ergodic means in ${\\rm End}\\, \\, C^{*} (\\Omega)$. We discuss conditions on the dynamical system under which (a) every ergodic net contains a convergent subsequence; (b) all ergodic nets converge; (c) all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on $\\Omega$, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup, the K\\\"{o}hler operator semigroup $\\Gamma $, and the semigroup $G$ that is the weak$^{*} $ closure of the convex hull of $\\Gamma $ in ${\\rm End}\\, C^{*} (\\Omega)$. The main results are stated for ordinary semicascades (whose Ellis semigroup is metrizable) and tame semicascades. For a dynamics, being ordinary is equivalent to being \"nonchaotic\" in an appropriate sense. We present a classification of compact dynamical systems in terms of topological properties of the above-mentioned semigroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-24T18:33:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "20M20"
    ],
    "keywords": [
      "discrete dynamical systems",
      "enveloping semigroup",
      "ergodic properties",
      "semicascade",
      "generalized operator ergodic means"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.6283R"
    }
  }
}