{
  "id": "1211.4541",
  "version": "v1",
  "published": "2012-11-19T19:31:53.000Z",
  "updated": "2012-11-19T19:31:53.000Z",
  "title": "On the derivative of the α-Farey-Minkowski function",
  "authors": [
    "Sara Munday"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we study the family of $\\alpha$-Farey-Minkowski functions $\\theta_\\alpha$, for an arbitrary countable partition $\\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\\alpha$-Farey systems and the tent map. We first show that each function $\\theta_\\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\\Theta_0:={x\\in\\U:\\theta_\\alpha'(x)=0}, \\Theta_\\infty:={x\\in\\U:\\theta_\\alpha'(x)=\\infty} and \\Theta_\\sim:=\\U\\setminus(\\Theta_0\\cup\\Theta_\\infty)$. The main result is that [\\dim_{\\mathrm{H}}(\\Theta_\\infty)=\\dim_{\\mathrm{H}}(\\Theta_\\sim)=\\sigma_\\alpha(\\log2)<\\dim_{\\mathrm{H}}(\\Theta_0)=1,] where $\\sigma_\\alpha(\\log2)$ is the Hausdorff dimension of the level set ${x\\in \\U:\\Lambda(F_\\alpha, x)=s}$, where $\\Lambda(F_\\alpha, x)$ is the Lyapunov exponent of the map $F_\\alpha$ at the point $x$. The proof of the theorem employs the multifractal formalism for $\\alpha$-Farey systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-19T19:31:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit interval",
      "farey systems",
      "derivative",
      "arbitrary countable partition",
      "tent map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4541M"
    }
  }
}