{
  "id": "2005.07451",
  "version": "v1",
  "published": "2020-05-15T10:01:09.000Z",
  "updated": "2020-05-15T10:01:09.000Z",
  "title": "Lipschitz classification of Bedford-McMullen carpets (I): Invariance of Multifractal spectrum and arithmetic doubling property",
  "authors": [
    "Hui Rao",
    "Ya-min Yang",
    "Yuan Zhang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $\\mu_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum of $\\mu_E$ is a bi-Lipschitz invariant, and the doubling property of $\\mu_E$ is also invariant under a bi-Lipschtz map. We show that if $E$ and $F$ are totally disconnected and that $\\mu_E$ and $\\mu_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a measure preserving property. Thanks to the above results, we give a complete classification of Bedford-McMullen carpets which are regular(that is, its Hausdorff dimension and box dimension coincides,) or satisfy a separation condition due to [J. F. King, The Singularity spectrum for general sierpinski carpets, \\textit{Adv. Math.} \\textbf{116} (1995), 1-11].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-15T10:01:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "26A16"
    ],
    "keywords": [
      "bedford-mcmullen carpets",
      "arithmetic doubling property",
      "multifractal spectrum",
      "invariance",
      "general sierpinski carpets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}