{
  "id": "1608.05535",
  "version": "v1",
  "published": "2016-08-19T08:47:19.000Z",
  "updated": "2016-08-19T08:47:19.000Z",
  "title": "Recurrence in the dynamical system $(X,\\langle T_s\\rangle_{s\\in S})$ and ideals of $βS$",
  "authors": [
    "Neil Hindman",
    "Dona Strauss",
    "Luca Q. Zamboni"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "A {\\it dynamical system\\/} is a pair $(X,\\langle T_s\\rangle_{s\\in S})$, where $X$ is a compact Hausdorff space, $S$ is a semigroup, for each $s\\in S$, $T_s$ is a continuous function from $X$ to $X$, and for all $s,t\\in S$, $T_s\\circ T_t=T_{st}$. Given a point $p\\in\\beta S$, the Stone-\\v Cech compactification of the discrete space $S$, $T_p:X\\to X$ is defined by, for $x\\in X$, $\\displaystyle T_p(x)=p{-}\\!\\lim_{s\\in S}T_s(x)$. We let $\\beta S$ have the operation extending the operation of $S$ such that $\\beta S$ is a right topological semigroup and multiplication on the left by any point of $S$ is continuous. Given $p,q\\in\\beta S$, $T_p\\circ T_q=T_{pq}$, but $T_p$ is usually not continuous. Given a dynamical system $(X,\\langle T_s\\rangle_{s\\in S})$, and a point $x\\in X$, we let $U(x)=\\{p\\in\\beta S:T_p(x)$ is uniformly recurrent$\\}$. We show that each $U(x)$ is a left ideal of $\\beta S$ and for any semigroup we can get a dynamical system with respect to which $K(\\beta S)=\\bigcap_{x\\in X}U(x)$ and $c\\ell K(\\beta S)=\\bigcap\\{U(x):x\\in X$ and $U(x)$ is closed$\\}$. And we show that weak cancellation assumptions guarantee that each such $U(x)$ properly contains $K(\\beta S)$ and has $U(x) \\setminus c\\ell K(\\beta S)\\neq \\emptyset$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-19T08:47:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dynamical system",
      "weak cancellation assumptions guarantee",
      "recurrence",
      "compact hausdorff space",
      "right topological semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}