{
  "id": "2104.06997",
  "version": "v1",
  "published": "2021-04-14T17:28:50.000Z",
  "updated": "2021-04-14T17:28:50.000Z",
  "title": "A Multifractal Decomposition for Self-similar Measures with Exact Overlaps",
  "authors": [
    "Alex Rutar"
  ],
  "comment": "60 pages, 5 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study self-similar measures in $\\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $\\mu$, we show that there is a finite set of concave functions $\\{\\tau_1,\\ldots,\\tau_m\\}$ such that the $L^q$-spectrum of $\\mu$ is given by $\\min\\{\\tau_1,\\ldots,\\tau_m\\}$ and the multifractal spectrum of $\\mu$ is given by $\\max\\{\\tau_1^*,\\ldots,\\tau_m^*\\}$, where $\\tau_i^*$ denotes the concave conjugate of $\\tau_i$. In particular, the measure $\\mu$ satisfies the multifractal formalism if and only if its multifractal spectrum is a concave function. This implies that $\\mu$ satisfies the multifractal formalism at values corresponding to points of differentiability of the $L^q$-spectrum. We also verify existence of the limit for the $L^q$-spectra of such measures for every $q\\in\\mathbb{R}$. As a direct application, we obtain many new results and simple proofs of well-known results in the multifractal analysis of self-similar measures satisfying the weak separation condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-14T17:28:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80"
    ],
    "keywords": [
      "exact overlaps",
      "multifractal decomposition",
      "weak separation condition",
      "multifractal formalism",
      "multifractal spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}