{
  "id": "1002.2912",
  "version": "v1",
  "published": "2010-02-15T16:46:59.000Z",
  "updated": "2010-02-15T16:46:59.000Z",
  "title": "Multifractal analysis and localized asymptotic behavior for almost additive potentials",
  "authors": [
    "Julien Barral",
    "Yan-Hui Qu"
  ],
  "comment": "40 pages, 3 figures",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost-additive continuous potentials $(\\phi_n)_{n=1}^\\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without bounded distorsion property assumption. We express the whole Hausdorff spectrum in terms of a conditional variational principle, as well as a new large deviations principle. Our approach provides a new description of the structure of the spectrum in terms of {\\it weak} concavity. Another new point is that we consider sets of points at which the asymptotic behavior of $\\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\\{x\\in X: \\lim_{n\\to\\infty} \\phi_n(x)/n=\\xi(x)\\}$, where $\\xi$ is a given continuous function. This is naturally related to Birkhoff's ergodic theorem and has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\\R^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-15T16:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37D35"
    ],
    "keywords": [
      "localized asymptotic behavior",
      "multifractal analysis",
      "additive potentials",
      "conformal planar cantor sets",
      "fine local behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.2912B"
    }
  }
}