{
  "id": "1307.5223",
  "version": "v1",
  "published": "2013-07-19T14:09:36.000Z",
  "updated": "2013-07-19T14:09:36.000Z",
  "title": "Multifractal tubes",
  "authors": [
    "Lars Olsen"
  ],
  "comment": "122 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Tube formulas refer to the study of volumes of $r$ neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a systematic study of tube formulas for fractal sets. Following this, it is natural to ask to what extend it is possible to develop a theory of multifractal tube formulas for multifractal measures. In this paper we propose and develop a framework for such a theory. Firstly, we de?ne multifractal tube formulas and, more generally, multifractal tube measures for general multifractal measures. Secondly, we introduce and develop two approaches for analysing these concepts for self-similar multifractal measures, namely: (1) Multifractal tubes of self-similar measures and renewal theory. Using techniques from renewal theory we give a complete description of the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures satisfying the Open Set Condition. (2) Multifractal tubes of self-similar measures and zeta-functions. Unfortunately, renewal theory techniques do not yield \"explicit\" expressions for the functions describing the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures. This is clearly undesirable. For this reason, we introduce and develop a second framework for studying multifractal tube formulas of self-similar measures. This approach is based on multifractal zeta-functions and allow us obtain \"explicit\" expressions for the multifractal tube formulas of self-similar measures, namely, using the Mellin transform and the residue theorem, we are able to express the multifractal tube formulas as sums involving the residues of the zeta-function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-19T14:09:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "37C30"
    ],
    "keywords": [
      "self-similar measures",
      "renewal theory",
      "asymptotic behaviour",
      "ne multifractal tube formulas",
      "tube formulas refer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 122,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5223O"
    }
  }
}