{
  "id": "1611.08290",
  "version": "v1",
  "published": "2016-11-24T19:18:31.000Z",
  "updated": "2016-11-24T19:18:31.000Z",
  "title": "Cardinality of the Ellis semigroup on compact metric countable spaces",
  "authors": [
    "S. Garcia-Ferreira",
    "Y. Rodriguez-Lopez",
    "C. Uzcategui"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \\setminus \\{ f^n : n \\in \\mathbb{N}\\}$ are both finite is given. We show that if the collection of all periods of the periodic points of $(X,f)$ is infinite, then $E(X,f)$ has size $2^{\\aleph_0}$. It is also proved that if $(X,f)$ has a point with a dense orbit and all elements of $E(X,f)$ are continuous, then $|E(X,f)| \\leq |X|$. For dynamical systems of the form $(\\omega^2 +1,f)$, we show that if there is a point with a dense orbit, then all elements of $E(\\omega^2+1,f)$ are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where $E(\\omega^2+1,f)$ and $\\omega^2+1$ are homeomorphic but not algebraically homeomorphic, where $\\omega^2+1$ is taken with the usual ordinal addition as semigroup operation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-24T19:18:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H20",
      "54G20",
      "54D80"
    ],
    "keywords": [
      "compact metric countable spaces",
      "ellis semigroup",
      "dense orbit",
      "cardinality",
      "dynamical system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}