{
  "id": "2103.10835",
  "version": "v1",
  "published": "2021-03-19T14:34:32.000Z",
  "updated": "2021-03-19T14:34:32.000Z",
  "title": "Topological mild mixing of all orders along polynomials",
  "authors": [
    "Yang Cao",
    "Song Shao"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open subsets $U,V$, $\\{n\\in \\Z: U\\cap T^{-n}V\\neq \\emptyset\\}$ is an IP$^*$-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that $(X,T)$ is a topologically mildly mixing minimal system, $d\\in \\N$, $p_1(n),\\ldots, p_d(n)$ are integral polynomials with no $p_i$ and no $p_i-p_j$ constant, $1\\le i\\neq j\\le d$, then for all non-empty open subsets $U , V_1, \\ldots, V_d $, $\\{n\\in \\Z: U\\cap T^{-p_1(n) }V_1\\cap T^{-p_2(n)}V_2\\cap \\ldots \\cap T^{-p_d(n) }V_d \\neq \\emptyset \\}$ is an IP$^*$-set. We also give the theorem for systems under abelian group actions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-19T14:34:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological mild mixing",
      "polynomials",
      "non-empty open subsets",
      "abelian group actions",
      "topologically mildly mixing minimal system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}