{
  "id": "2405.10179",
  "version": "v1",
  "published": "2024-05-16T15:21:47.000Z",
  "updated": "2024-05-16T15:21:47.000Z",
  "title": "Conditions on the continuity of the Hausdorff measure",
  "authors": [
    "Rafał Tryniecki"
  ],
  "comment": "25 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $b_k$ be strictly decreasing sequence of real numbers such that $b_0 = 1$ and $f_k$ be decreasing, linear functions such that $f_k(b_k) = 1$ and $f_k(b_{k-1}) = 0$, $k = 1, 2, \\dots$. We define iterated function system (IFS) $S_n$ by limiting the collection of functions $f_k$ to first n, meaning $S_n = \\{f_k \\}_{k=1}^n$. Let $J_n$ denote the limit set of $S_n$. We show that if $S_n$ fulfills the following two conditions: (1)~$\\lim\\limits_{n \\to \\infty} \\left(1-h_n\\right) \\ln{n} = 0 $ where $h_n$ is the Hausdorff dimension of $J_n$, and (2)~$\\sup \\limits_{k\\in \\mathbb{N}} \\left \\{\\frac{b_k-b_{k+1}}{b_{k+1}} \\right \\} < \\infty $, then $\\lim\\limits_{n\\to \\infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. We also show examples of families of IFSes fulfilling those properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-16T15:21:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05"
    ],
    "keywords": [
      "conditions",
      "hausdorff dimension",
      "continuity",
      "define iterated function system",
      "corresponding hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}