{
  "id": "0802.3433",
  "version": "v2",
  "published": "2008-02-23T09:09:27.000Z",
  "updated": "2008-04-23T16:40:34.000Z",
  "title": "On Khintchine exponents and Lyapunov exponents of continued fractions",
  "authors": [
    "Ai-Hua Fan",
    "Ling-Min Liao",
    "Bao-Wei Wang",
    "Jun Wu"
  ],
  "comment": "37 pages, 5 figures, accepted by Ergodic Theory and Dyanmical Systems",
  "categories": [
    "math.DS"
  ],
  "abstract": "Assume that $x\\in [0,1) $ admits its continued fraction expansion $x=[a_1(x), a_2(x),...]$. The Khintchine exponent $\\gamma(x)$ of $x$ is defined by $\\gamma(x):=\\lim\\limits_{n\\to \\infty}\\frac{1}{n}\\sum_{j=1}^n \\log a_j(x)$ when the limit exists. Khintchine spectrum $\\dim E_\\xi$ is fully studied, where $ E_{\\xi}:=\\{x\\in [0,1):\\gamma(x)=\\xi\\} (\\xi \\geq 0)$ and $\\dim$ denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum $\\dim E_{\\xi}$, as function of $\\xi \\in [0, +\\infty)$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\\gamma^{\\phi}(x):=\\lim\\limits_{n\\to\\infty}\\frac{1}{\\phi(n)} \\sum_{j=1}^n \\log a_j(x)$ are also studied, where $\\phi (n)$ tends to the infinity faster than $n$ does. Under some regular conditions on $\\phi$, it is proved that the fast Khintchine spectrum $\\dim (\\{x\\in [0,1]: \\gamma^{\\phi}(x)= \\xi \\}) $ is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-23T16:40:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A78",
      "28A80"
    ],
    "keywords": [
      "lyapunov exponents",
      "fast lyapunov spectrum",
      "fast khintchine spectrum",
      "continued fraction expansion",
      "usual point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.3433F"
    }
  }
}