{
  "id": "1807.10155",
  "version": "v1",
  "published": "2018-07-25T15:47:47.000Z",
  "updated": "2018-07-25T15:47:47.000Z",
  "title": "An answer of Furstenberg' problem on topological disjointness for transitive systems",
  "authors": [
    "Wen Huang",
    "Song Shao",
    "Xiangdong Ye"
  ],
  "comment": "15 pages. arXiv admin note: text overlap with arXiv:0910.3362",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we give an answer of Furstenberg' problem on topological disjointness for transitive systems. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\\in Y$ and any open neighbourhood $V$ of $y$, and for any nonempty open subset $U\\subset X$, there is $x\\in D\\cap U$ satisfying that $\\{n\\in { \\mathbb Z}_+: T^n x\\in U, S^n y\\in V\\}$ is syndetic. As applications we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^n,T^{(n)})$ and $(X, T^n)$ for any $n\\in {\\mathbb N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_K)$ is disjoint from all minimal systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-25T15:47:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "54H20"
    ],
    "keywords": [
      "transitive system",
      "minimal system",
      "topological disjointness",
      "furstenberg",
      "nonempty open subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}