{
  "id": "2106.12418",
  "version": "v1",
  "published": "2021-06-23T14:11:27.000Z",
  "updated": "2021-06-23T14:11:27.000Z",
  "title": "On Limit sets of Monotone maps on Regular curves",
  "authors": [
    "Aymen Daghar",
    "Habib Marzougui"
  ],
  "comment": "18 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate the structure of $\\omega$-limit (resp. $\\alpha$-limit) sets for a monotone map $f$ on a regular curve $X$. %Let $X$ be a regular curve and let $f: X\\longrightarrowX$ be a monotone map. We show that for any $x\\in X$ (resp. for any negative orbit $(x_{n})_{n\\geq 0}$ of $x$), the $\\omega$-limit set $\\omega_{f}(x)$ (resp. $\\alpha$-limit set $\\alpha_{f}((x_{n})_{n\\geq 0})$) is a minimal set. This also hold for $\\alpha$-limit set $\\alpha_{f}(x)$ whenever $x$ is not a periodic point. These results extend those of Naghmouchi \\cite{n} %[J. Difference Equ. Appl., 23 (2017), 1485--1490] established whenever $f$ is a homeomorphism on a regular curve and those of Abdelli \\cite{a} %[Chaos, Solitons Fractals, 71 (2015), 66--72] , whenever $f$ is a monotone map on a local dendrite. Further results related to the basin of attraction of an infinite minimal set are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-23T14:11:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B20",
      "37B45",
      "54H20"
    ],
    "keywords": [
      "monotone map",
      "regular curve",
      "limit set",
      "infinite minimal set",
      "results extend"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}