{
  "id": "2308.02639",
  "version": "v1",
  "published": "2023-08-04T18:00:35.000Z",
  "updated": "2023-08-04T18:00:35.000Z",
  "title": "Lipschitz images and dimensions",
  "authors": [
    "Richárd Balka",
    "Tamás Keleti"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that if $A$ and $B$ are compact metric spaces and the Hausdorff dimension of $A$ is bigger than the upper box dimension of $B$, then there exist a compact set $A'\\subset A$ and a Lipschitz onto map $f\\colon A'\\to B$. As a corollary we prove that any `natural' dimension in $\\mathbb{R}^n$ must be between the Hausdorff and upper box dimensions. We show that if $A$ and $B$ are self-similar sets with the strong separation condition with equal Hausdorff dimension and $A$ is homogeneous, then $A$ can be mapped onto $B$ by a Lipschitz map if and only if $A$ and $B$ are bilipschitz equivalent. For given $\\alpha>0$ we also give a characterization of those compact metric spaces that can be obtained as an $\\alpha$-H\\\"older image of a compact subset of $\\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-04T18:00:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80",
      "51F30",
      "54E45"
    ],
    "keywords": [
      "lipschitz image",
      "compact metric space",
      "upper box dimension",
      "middle third cantor set",
      "equal hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}