{
  "abstract": "For a complex single variable polynomial $f$ of degree $d$, let $K$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K$ has an invariant component $K^*$ on which $f$ acts as a degree $d_*<d$ map. This is a simplest instance of holomorphic polynomial-like renormalization (Douady-Hubbard). One can associate a certain Cantor-like subset $G'$ of the circle with $K^*$; it is defined as the set of arguments of all smooth or broken rays to $K^*$. We will describe a role the Hausdorff dimension of $G'$ and the respective Hausdorff measure play in geometry of $K^*$. In particular, we give upper and lower bounds on the modulus of renormalization in terms of the Hausdorff measure of $K^*$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.20188",
  "author_total": 3,
  "authors": [
    {
      "name": "Alexander Blokh"
    },
    {
      "name": "Lex Oversteegen"
    },
    {
      "name": "Vladlen Timorin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2m5p97p6k2nhs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.20188",
  "fulltext_url": null,
  "id": "2m5p97p6k2nhs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.20188",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7165367094",
  "pdf_url": "https://arxiv.org/pdf/2606.20188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2m5p97p6k2nhs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2606.20188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.20188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Renormalization, equipotential annuli and the Hausdorff measure",
  "title_text": "Renormalization, equipotential annuli and the Hausdorff measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/2m5p97p6k2nhs",
  "venue": "arXiv",
  "versions": []
}