{
  "id": "1103.3412",
  "version": "v2",
  "published": "2011-03-17T13:50:45.000Z",
  "updated": "2011-07-30T12:24:31.000Z",
  "title": "Transitive points via Furstenberg family",
  "authors": [
    "Jian Li"
  ],
  "comment": "Minor changes, 19 pages,to appear in Topology and its applications",
  "journal": "Topology and its Applications 158 (2011) pp. 2221-2231",
  "doi": "10.1016/j.topol.2011.07.013",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X,T)$ be a topological dynamical system and $\\mathcal{F}$ be a Furstenberg family (a collection of subsets of $\\mathbb{Z}_+$ with hereditary upward property). A point $x\\in X$ is called an $\\mathcal{F}$-transitive one if $\\{n\\in\\mathbb{Z}_+:\\, T^n x\\in U\\}\\in\\F$ for every nonempty open subset $U$ of $X$; the system $(X,T)$ is called $\\F$-point transitive if there exists some $\\mathcal{F}$-transitive point. In this paper, we aim to classify transitive systems by $\\mathcal{F}$-point transitivity. Among other things, it is shown that $(X,T)$ is a weakly mixing E-system (resp.\\@ weakly mixing M-system, HY-system) if and only if it is $\\{\\textrm{D-sets}\\}$-point transitive (resp.\\@ $\\{\\textrm{central sets}\\}$-point transitive, $\\{\\textrm{weakly thick sets}\\}$-point transitive). It is shown that every weakly mixing system is $\\mathcal{F}_{ip}$-point transitive, while we construct an $\\mathcal{F}_{ip}$-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is $\\Delta^*(\\mathcal{F}_{wt})$-transitive if and only if it is weakly disjoint from every P-system.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-07-30T12:24:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37B40",
      "54H20"
    ],
    "keywords": [
      "transitive point",
      "furstenberg family",
      "weakly mixing",
      "dense small periodic sets",
      "nonempty open subset"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3412L"
    }
  }
}