{
  "id": "math/0601564",
  "version": "v2",
  "published": "2006-01-23T18:21:17.000Z",
  "updated": "2006-08-08T16:58:30.000Z",
  "title": "Distortion bounds for $C^{2+η}$ unimodal maps",
  "authors": [
    "Mike Todd"
  ],
  "comment": "5 figures. Typos corrected, and some clarifications made; principally to the first part of Section 4. To appear in Fundamenta Mathematicae",
  "journal": "Fund Math 193 (2007) 37-77",
  "categories": [
    "math.DS"
  ],
  "abstract": "We obtain estimates for derivative and cross--ratio distortion for $C^{2+\\eta}$ (any $\\eta>0$) unimodal maps with non--flat critical points. We do not require any `Schwarzian--like' condition. For two intervals $J \\subset T$, the cross--ratio is defined as the value $$B(T,J):=\\frac{|T||J|}{|L||R|}$$ where $L,R$ are the left and right connected components of $T\\setminus J$ respectively. For an interval map $g$ \\st $g_T:T \\to \\RRR$ is a diffeomorphism, we consider the cross--ratio distortion to be $$B(g,T, J):=\\frac{B(g(T),g(J))}{B(T,J)}.$$ We prove that for all $0<K<1$ there exists some interval $I_0$ around the critical point \\st for any intervals $J \\subset T$, if $f^n|_T$ is a diffeomorphism and $f^n(T) \\subset I_0$ then $$B(f^n, T, J)> K.$$ Then the distortion of derivatives of $f^n|_J$ can be estimated with the Koebe Lemma in terms of $K$ and $B(f^n(T),f^n(J))$. This tool is commonly used to study topological, geometric and ergodic properties of $f$. This extends a result of Kozlovski.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-08-08T16:58:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05"
    ],
    "keywords": [
      "unimodal maps",
      "distortion bounds",
      "cross-ratio distortion",
      "interval map",
      "diffeomorphism"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1564T"
    }
  }
}