{
  "id": "1110.6125",
  "version": "v2",
  "published": "2011-10-27T15:51:37.000Z",
  "updated": "2012-10-04T16:39:38.000Z",
  "title": "On conjugations of circle homeomorphisms with two break points",
  "authors": [
    "Habibulla Akhadkulov",
    "Akhtam Dzhalilov",
    "Dieter Mayer"
  ],
  "comment": "16 pages, 2 figures, to appear in Ergodic Theory and Dynamical Systems",
  "doi": "10.1017/etds.2012.159",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f_i\\in C^{2+\\alpha}(S^1\\setminus \\{a_i,b_i\\}), \\alpha >0, i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and $\\mu_1([a_1,b_1])= \\mu_2([a_2,b_2])$, where $\\mu_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\\times \\frac{Df_1(b_1-0)}{Df_1(b_1+0)}\\neq \\frac{Df_2(a_2-0)}{Df_2(a_2+0)}\\times \\frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $\\psi$ conjugating $f_1$ and $f_2$ is a singular function, i.e. it is continuous on $S^1$, but $D\\psi = 0$ a.e. with respect to Lebesgue measure",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-04T16:39:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E10",
      "37C15",
      "37C40"
    ],
    "keywords": [
      "circle homeomorphisms",
      "break points",
      "conjugations",
      "identical irrational rotation number",
      "invariant measures"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6125A"
    }
  }
}