{
  "id": "1307.4138",
  "version": "v3",
  "published": "2013-07-16T00:57:38.000Z",
  "updated": "2015-07-04T07:47:29.000Z",
  "title": "Point transitivity, $Δ$-transitivity and multi-minimality",
  "authors": [
    "Zhijing Chen",
    "Jian Li",
    "Jie Lü"
  ],
  "comment": "19 pages, the final version. Change the title to \"Point transitivity, $\\Delta$-transitivity and multi-minimality\"",
  "journal": "Ergodic Theory and Dynamical Systems, 35 (2015), no. 5, 1423--1442",
  "doi": "10.1017/etds.2013.106",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X, f)$ be a topological dynamical system and $\\mathcal {F}$ be a Furstenberg family (a collection of subsets of $\\mathbb{N}$ with hereditary upward property). A point $x\\in X$ is called an $\\mathcal {F}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\\{n\\in \\mathbb{N}: f^{n}(x) \\in U\\}$, is in $\\mathcal {F}$; the system $(X,f)$ is called $\\mathcal {F}$-point transitive if there exists some $\\mathcal {F}$-transitive point. In this paper, we first discuss the connection between $\\mathcal {F}$-point transitivity and $\\mathcal {F}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by $\\mathcal {F}$-point transitivity, completing results in [Transitive points via Furstenberg family, Topology Appl. 158 (2011), 2221--2231]. We also show that multi-transitivity, $\\Delta$-transitivity and multi-minimality can also be characterized by $\\mathcal {F}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661--1672].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-22T13:49:10.000Z",
      "title": "Point transitivity, Delta-transitivity and multi-minimality",
      "comment": "19papger, changes suggested by the referee incorporated, to appear in Ergodic Theory and Dynamical Systems. arXiv admin note: text overlap with arXiv:1307.3817",
      "journal": null
    },
    {
      "version": "v3",
      "updated": "2015-07-04T07:47:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H20",
      "37B40",
      "58K15",
      "37B45"
    ],
    "keywords": [
      "point transitivity",
      "multi-minimality",
      "transitive point",
      "delta-transitivity",
      "non-empty open subset"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.4138C"
    }
  }
}