{
  "id": "1706.03654",
  "version": "v1",
  "published": "2017-06-12T14:21:39.000Z",
  "updated": "2017-06-12T14:21:39.000Z",
  "title": "Renormalizations of circle maps with several break points",
  "authors": [
    "Kleyber Cunha",
    "Akhtam Dzhalilov",
    "Abdumajid Begmatov"
  ],
  "comment": "28 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f$ be an orientation preserving homeomorphisms on the circle with several break points, that is, its derivative $Df$ has jump discontinuities at these points. We study Rauzy-Veech renormalizations of piecewise smooth circle homeomorphisms, by considering such maps as generalized interval exchange maps with genus one. Suppose that $Df$ is absolutely continuous on the each interval of continuity and $D\\ln{Df}\\in \\mathbb{L}_{p}$ for some $p>1$. We prove that, under certain combinatorial assumptions on $f$, renormalizations $R^{n}(f)$ are approximated by piecewise M\\\"{o}bus functions in $C^{1+L_{1}}$-norm, that means, $R^{n}(f)$ are approximated in $C^{1}$-norm and $D^{2}R^{n}(f)$ are approximated in $L_{1}$-norm. In particular, if $f$ has trivial product of size of breaks, then the renormalizations are approximated by piecewise affine interval exchange maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-12T14:21:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E10",
      "37E05",
      "37E20",
      "37C05",
      "37B10"
    ],
    "keywords": [
      "break points",
      "circle maps",
      "piecewise affine interval exchange maps",
      "piecewise smooth circle homeomorphisms",
      "study rauzy-veech renormalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}