{
  "id": "math/0408047",
  "version": "v1",
  "published": "2004-08-03T18:15:15.000Z",
  "updated": "2004-08-03T18:15:15.000Z",
  "title": "A Modified Multifractal Formalism for a Class of Self-Similar Measures",
  "authors": [
    "Pablo Shmerkin"
  ],
  "journal": "Asian J. Math. 9 (2005), no. 3, 323--348",
  "categories": [
    "math.CA"
  ],
  "abstract": "The multifractal spectrum of a Borel measure $\\mu$ in $\\mathbb{R}^n$ is defined as \\[ f_\\mu(\\alpha) = \\dim_H {x:\\lim_{r\\to 0} \\frac{\\log \\mu(B(x,r))}{\\log r}=\\alpha}. \\] For self-similar measures under the open set condition the behavior of this and related functions is well-understood; the situation turns out to be very regular and is governed by the so-called ''multifractal formalism''. Recently there has been a lot of interest in understanding how much of the theory carries over to the overlapping case; however, much less is known in this case and what is known makes it clear that more complicated phenomena are possible. Here we carry out a complete study of the multifractal structure for a class of self-similar measures with overlap which includes the 3-fold convolution of the Cantor measure. Among other things, we prove that the multifractal formalism fails for many of these measures, but it holds when taking a suitable restriction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-08-03T18:15:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78"
    ],
    "keywords": [
      "self-similar measures",
      "modified multifractal formalism",
      "open set condition",
      "multifractal formalism fails",
      "multifractal spectrum"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8047S"
    }
  }
}