{
  "id": "1110.6599",
  "version": "v1",
  "published": "2011-10-30T10:42:25.000Z",
  "updated": "2011-10-30T10:42:25.000Z",
  "title": "Higher order almost automorphy, recurrence sets and the regionally proximal relation",
  "authors": [
    "Wen Huang",
    "Song Shao",
    "Xiangdong Ye"
  ],
  "comment": "37 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, $d$-step almost automorphic systems are studied for $d\\in\\N$, which are the generalization of the classical almost automorphic ones. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\\in X$ is $d$-step almost automorphic can be characterized via various subsets of $\\Z$ including the dual sets of $d$-step Poincar\\'e and Birkhoff recurrence sets, and Nil$_d$ Bohr$_0$-sets by considering $N(x,V)=\\{n\\in\\Z: T^nx\\in V\\}$, where $V$ is an arbitrary neighborhood of $x$. Moreover, it turns out that the condition $(x,y)\\in X\\times X$ is regionally proximal of order $d$ can also be characterized via various subsets of $\\Z$ including $d$-step Poincar\\'e and Birkhoff recurrence sets, $SG_d$ sets, the dual sets of Nil$_d$ Bohr$_0$-sets, and others by considering $N(x,U)=\\{n\\in\\Z: T^nx\\in U\\}$, where $U$ is an arbitrary neighborhood of $y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-30T10:42:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37B20"
    ],
    "keywords": [
      "regionally proximal relation",
      "higher order",
      "birkhoff recurrence sets",
      "arbitrary neighborhood",
      "automorphy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6599H"
    }
  }
}