{
  "id": "1510.06663",
  "version": "v1",
  "published": "2015-10-22T15:54:11.000Z",
  "updated": "2015-10-22T15:54:11.000Z",
  "title": "On a generalization of the Cartwright-Littlewood fixed point theorem for planar homeomorphisms",
  "authors": [
    "Jan P. Boroński"
  ],
  "comment": "Accepted to Ergodic Theory and Dynamical Systems",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose $h : \\mathbb{R}^2 \\to\\mathbb{R}^2$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\\cup C$ is acyclic. If there is a $c\\in C$ such that $\\{h^{-i}(c):i\\in\\mathbb{N}\\}\\subseteq C$, or $\\{h^i(c):i\\in\\mathbb{N}\\}\\subseteq C$, then $C$ also contains a fixed point of $h$. Our approach is based on Morton Brown's short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a $2$-periodic orbit in $C$ if it contains a $k$-periodic orbit ($k>1$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-22T15:54:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E30",
      "37C25",
      "55C20",
      "54H25"
    ],
    "keywords": [
      "cartwright-littlewood fixed point theorem",
      "generalization",
      "morton browns short proof",
      "orientation preserving planar homeomorphism",
      "linked periodic orbits theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}