{
  "id": "1810.12854",
  "version": "v1",
  "published": "2018-10-30T16:53:52.000Z",
  "updated": "2018-10-30T16:53:52.000Z",
  "title": "Topological Dynamics of Enveloping Semigroups",
  "authors": [
    "Anima Nagar",
    "Manpreet Singh"
  ],
  "comment": "61 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "A compact metric space $X$ and a discrete topological acting group $T$ give a flow $(X,T)$. Robert Ellis had initiated the study of dynamical properties of the flow $(X,T)$ via the algebraic properties of its \"Enveloping Semigroup\" $E(X)$. This concept of \\emph{Enveloping Semigroups} that he defined, has turned out to be a very fundamental tool in the abstract theory of `topological dynamics'. The flow $(X,T)$ induces the flow $(2^X,T)$. Such a study was first initiated by Eli Glasner who studied the properties of this induced flow by defining and using the notion of a `circle operator' as an action of $\\beta T$ on $2^X$, where $\\beta T$ is the \\emph{Stone-$\\check{C}$ech compactification} of $T$ and also a universal enveloping semigroup. We propose that the study of properties for the induced flow $(2^X,T)$ be made using the algebraic properties of $E(2^X)$ on the lines of Ellis' \\ theory, instead of looking into the action of $\\beta T$ on $2^X$ via the circle operator as done by Glasner. Such a study requires extending the present theory on the flow $(E(X),T)$. In this article, we take up such a study giving some subtle relations between the semigroups $E(X)$ and $E(2^X)$ and some interesting associated consequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-30T16:53:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "54H20",
      "37B20",
      "54B20"
    ],
    "keywords": [
      "topological dynamics",
      "circle operator",
      "algebraic properties",
      "induced flow",
      "compact metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}