{
  "abstract": "For the quadratic family $f_c(z)=z^2+c$, we study the non-escaping locus $\\KM=\\{(c,z):c\\in\\M,\\ z\\in K_c\\}$, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here $\\M$ is the Mandelbrot set, $K_c$ is the filled Julia set of $f_c$, and $J_c=\\partial K_c$. We prove that $\\partial\\KM$ has full Hausdorff dimension $4$. The set $\\KM$ is locally connected at every $(c,z)$ with hyperbolic $c\\in\\M$. At parameters on $\\partial\\M$, we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of $\\KM$ at $(c,z)$. However, local connectedness fails at some $(c,z)$ with $c$ real parabolic and $z\\in K_c\\setminus J_c$, even though both the parameter space and the filled Julia fiber are locally connected there.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11793",
  "author_total": 1,
  "authors": [
    {
      "name": "Yueyang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6n6543yqxwyng.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11793",
  "fulltext_url": null,
  "id": "6n6543yqxwyng",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11793",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221115212",
  "pdf_url": "https://arxiv.org/pdf/2610.11793",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6n6543yqxwyng.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11793",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11793"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set",
  "title_text": "Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/6n6543yqxwyng",
  "venue": "arXiv",
  "versions": []
}