{
  "id": "2304.03097",
  "version": "v1",
  "published": "2023-04-06T14:24:56.000Z",
  "updated": "2023-04-06T14:24:56.000Z",
  "title": "Some dynamical properties related to polynomials",
  "authors": [
    "Qinqi Wu"
  ],
  "comment": "14 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $d\\in\\mathbb{Z}$ and $p_i$ be an integral polynomial with $p_i(0)=0,1\\leq i\\leq d$. It is shown that if $S$ is thickly syndetic in $\\mathbb{Z}$, then $\\{(m,n)\\in\\mathbb{Z}^2:m+p_i(n),m+p_2(n),\\ldots,m+p_d(n)\\in S\\}$ is thickly syndetic in $\\mathbb{Z}^2$. Meanwhile, we construct a transitive, strong mixing and non-minimal topological dynamical system $(X,T)$, such that the set $\\{x\\in X:\\forall\\ \\text{open}\\ U\\ni x,\\exists\\ n\\in\\mathbb{Z} \\ \\text{s.t.}\\ T^{n}x\\in U,T^{2n}x\\in U\\}$ is not dense in $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-06T14:24:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37B20"
    ],
    "keywords": [
      "dynamical properties",
      "thickly syndetic",
      "integral polynomial",
      "non-minimal topological dynamical system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}