{
  "id": "1210.3556",
  "version": "v1",
  "published": "2012-10-12T15:51:40.000Z",
  "updated": "2012-10-12T15:51:40.000Z",
  "title": "Displacement sequence of an orientation preserving circle homeomorphism",
  "authors": [
    "Wacław Marzantowicz",
    "Justyna Signerska"
  ],
  "comment": "18 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We give a complete description of the behaviour of the sequence of displacements $\\eta_n(z)=\\Phi^n(x) - \\Phi^{n-1}(x) \\ \\rmod \\ 1$, $z=\\exp(2\\pi \\rmi x)$, along a trajectory $\\{\\varphi^{n}(z)\\}$, where $\\varphi$ is an orientation preserving circle homeomorphism and $\\Phi:\\mathbb{R} \\to \\mathbb{R}$ its lift. If the rotation number $\\varrho(\\varphi)=\\frac{p}{q}$ is rational then $\\eta_n(z)$ is asymptotically periodic with semi-period $q$. This convergence to a periodic sequence is uniform in $z$ if we admit that some points are iterated backward instead of taking only forward iterations for all $z$. If $\\varrho(\\varphi) \\notin \\mathbb{Q}$ then the values of $\\eta_n(z)$ are dense in a set which depends on the map $\\gamma$ (semi-)conjugating $\\varphi$ with the rotation by $\\varrho(\\varphi)$ and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if $\\varphi$ is $C^1$-diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism $\\varphi$. Finally, we prove that even for the irrational rotation number $\\varrho$ the displacement sequence exhibits some regularity properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-12T15:51:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E10",
      "37E30",
      "37N25"
    ],
    "keywords": [
      "orientation preserving circle homeomorphism",
      "displacement sequence",
      "displacement distribution",
      "irrational rotation number",
      "periodic sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3556M"
    }
  }
}