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PreprintCC BY 4.0arXiv

Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set

  • Yueyang Wang
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Abstract

For the quadratic family fc(z)=z2+c, we study the non-escaping locus KM={(c,z):c∈M, z∈ Kc}, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here M is the Mandelbrot set, Kc is the filled Julia set of fc, and Jc=∂ Kc. We prove that ∂KM has full Hausdorff dimension $4$. The set KM is locally connected at every (c,z) with hyperbolic c∈M. At parameters on ∂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∈ Kc∖ Jc, even though both the parameter space and the filled Julia fiber are locally connected there.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026local,
  title = {{Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set}},
  author = {Yueyang Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11793},
  eprint = {2610.11793},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11793},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set
AU  - Yueyang Wang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11793
UR  - https://arxiv.org/abs/2610.11793
AB  - For the quadratic family f_c(z)=z^2+c, we study the non-escaping locus KM={(c,z):c∈M, z∈ 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=∂ K_c. We prove that ∂KM has full Hausdorff dimension $4$. The set KM is locally connected at every (c,z) with hyperbolic c∈M. At parameters on ∂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∈ K_c∖ J_c, even though both the parameter space and the filled Julia fiber are locally connected there.
ER  -