{
  "id": "1402.4756",
  "version": "v1",
  "published": "2014-02-19T18:19:42.000Z",
  "updated": "2014-02-19T18:19:42.000Z",
  "title": "Boundaries of the Arnol'd tongues and the standard family",
  "authors": [
    "Kuntal Banerjee"
  ],
  "comment": "30 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "For a family $(F_{t,a} : x \\mapsto x + t + a\\phi(x))$ of increasing homeomorphisms of $\\mathbb R$ with $\\phi$ being Lipschitz continuous of period 1, there is a parameter space consisting of the values $(t,a)$ such that the map $F_{t,a}$ is strictly increasing and it induces an orientation preserving circle homeomorphism. For each $\\theta \\in \\mathbb R$ there is an \\textsf{Arnol'd tongue} $\\mathcal T_\\theta$ of \\textsf{translation number} $\\theta$ in the parameter space. Given a rational $p/q$, it is shown that the boundary $\\partial \\mathcal T_{p/q}$ is a union of two Lipschitz curves which intersect at $a=0$ and there can be a non zero angle between them. In this direction we compute the first order asymptotic expansion of the boundaries of the rational and irrational tongues in the parameter space around $a=0$. For the standard family $(S_{t,a} : x \\mapsto x + t + a \\sin(2\\pi x))$, the boundary curves of $\\mathcal T_{p/q}$ have the same tangency at $a=0$ for $q\\ge 2$ and it is known that $q$ is their \\textsf{order of contact}. Using the techniques of \\textsf{guided} and \\textsf{admissible family}, we give a new proof of this. In particular we relate this to the \\textsf{parabolic multiplicity} of the map $s_{p/q} : z \\mapsto e^{i2\\pi p/q}ze^{\\pi z}$ at $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-19T18:19:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E10",
      "26A18",
      "30D05"
    ],
    "keywords": [
      "arnold tongues",
      "standard family",
      "parameter space",
      "first order asymptotic expansion",
      "orientation preserving circle homeomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.4756B"
    }
  }
}