{
  "id": "2108.02454",
  "version": "v1",
  "published": "2021-08-05T08:40:21.000Z",
  "updated": "2021-08-05T08:40:21.000Z",
  "title": "On Cartwright-Littlewood Fixed Point Theorem",
  "authors": [
    "Przemysław Kucharski"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove the following generalization of the Cartwright-Littlewood fixed point theorem. Suppose $ h\\colon~{\\mathbb R}^{2}\\to{\\mathbb R}^{2} $ is an orientation preserving planar homeomorphism, and $ X $ is an acyclic continuum. Let $ C $ be a component of $ X \\cap h(X) $. If there is a $ c \\in C $ such that $ {\\mathcal O}_{+} (c) \\subseteq C $ or $ {\\mathcal O}_{-} (c) \\subseteq C $ then $ C $ also contains a fixed point of $ h$. Our result also generalizes earlier results of Ostrovski and Boro\\'nski, and answers the Question from Boro\\'nski's work in 2017. The proof is inspired by a short proof of the result of Cartwright and Littlewood due to Hamilton.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-05T08:40:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cartwright-littlewood fixed point theorem",
      "orientation preserving planar homeomorphism",
      "generalizes earlier results",
      "acyclic continuum",
      "short proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}