{
  "id": "1111.2414",
  "version": "v1",
  "published": "2011-11-10T08:16:54.000Z",
  "updated": "2011-11-10T08:16:54.000Z",
  "title": "Multifractal analysis of Bernoulli convolutions associated with Salem numbers",
  "authors": [
    "De-Jun Feng"
  ],
  "comment": "26 pages. Accepted by Adv. Math",
  "categories": [
    "math.CA",
    "math.DS",
    "math.NT"
  ],
  "abstract": "We consider the multifractal structure of the Bernoulli convolution $\\nu_{\\lambda}$, where $\\lambda^{-1}$ is a Salem number in $(1,2)$. Let $\\tau(q)$ denote the $L^q$ spectrum of $\\nu_\\lambda$. We show that if $\\alpha \\in [\\tau'(+\\infty), \\tau'(0+)]$, then the level set $$E(\\alpha):={x\\in \\R:\\; \\lim_{r\\to 0}\\frac{\\log \\nu_\\lambda([x-r, x+r])}{\\log r}=\\alpha}$$ is non-empty and $\\dim_HE(\\alpha)=\\tau^*(\\alpha)$, where $\\tau^*$ denotes the Legendre transform of $\\tau$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[\\tau'(+\\infty), \\tau'(0+)]$ is not a singleton when $\\lambda^{-1}$ is the largest real root of the polynomial $x^{n}-x^{n-1}-... -x+1$, $n\\geq 4$. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-10T08:16:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80",
      "11K16"
    ],
    "keywords": [
      "bernoulli convolutions",
      "salem number",
      "multifractal analysis",
      "rich multifractal structures",
      "asymptotically weak separation condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.2414F"
    }
  }
}