{
  "id": "2002.02319",
  "version": "v1",
  "published": "2020-02-06T15:50:29.000Z",
  "updated": "2020-02-06T15:50:29.000Z",
  "title": "On multifractal formalism for self-similar measures with overlaps",
  "authors": [
    "Julien Barral",
    "De-Jun Feng"
  ],
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Let $\\mu$ be a self-similar measure generated by an IFS $\\Phi=\\{\\phi_i\\}_{i=1}^\\ell$ of similarities on $\\mathbb R^d$ ($d\\ge 1$). When $\\Phi$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $\\tau_\\mu(q)$ of $\\mu$ over $[0,1]$, and show that $\\tau_\\mu$ is differentiable over $(0,1]$ and the multifractal formalism holds for $\\mu$ at any $\\alpha\\in [\\tau_\\mu'(1),\\tau_\\mu'(0+)]$. We also verify the validity of the multifractal formalism of $\\mu$ over $[\\tau_\\mu'(\\infty),\\tau_\\mu'(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to a recent result of Shmerkin on the $L^q$-spectrum of self-similar measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-06T15:50:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37C45"
    ],
    "keywords": [
      "self-similar measure",
      "asymptotically weak separation condition holds",
      "multifractal formalism holds",
      "proof appeals",
      "dimensional regular"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}