{
  "id": "2502.02778",
  "version": "v1",
  "published": "2025-02-04T23:41:13.000Z",
  "updated": "2025-02-04T23:41:13.000Z",
  "title": "The hyperspace ω(f) when f is a transitive dendrite mapping",
  "authors": [
    "Jorge M. Martínez-Montejano",
    "Héctor Méndez",
    "Yajaida N. Velázquez-Inzunza"
  ],
  "categories": [
    "math.GN",
    "math.DS"
  ],
  "abstract": "Let $X$ be a compact metric space. By $2^X$ we denote the hyperspace of all closed and non-empty subsets of $X$ endowed with the Hausdorff metric. Let $f:X\\to X$ be a continuous function. In this paper we study some topological properties of the hyperspace $\\omega(f)$, the collection of all omega limits sets $\\omega(x,f)$ with $x\\in X$. We prove the following: $i)$ If $X$ has no isolated points, then, for every continuous function $f:X\\to X$, $int_{2^X}(\\omega(f))=\\emptyset$. $ii)$ If $X$ is a dendrite for which every arc contains a free arc and $f:X\\to X$ is transitive, then the hyperspace $\\omega(f)$ is totally disconnected. $iii)$ Let $D_\\infty$ be the Wazewski's universal dendrite. Then there exists a transitive continuous function $f:D_\\infty\\to D_\\infty$ for which the hyperspace $\\omega(f)$ contains an arc; hence, $\\omega(f)$ is not totally disconnected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-04T23:41:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "transitive dendrite mapping",
      "hyperspace",
      "continuous function",
      "omega limits sets",
      "wazewskis universal dendrite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}