Research Problems in Function Theory — Problem 2.88

Definitions and mathematical statement

For $c\in\mathbb C$, write $P_c(z)=z^2+c$. The Mandelbrot set is \[ \mathcal M=\{c\in\mathbb C:\sup_{n\ge0}|P_c^n(0)|<\infty\}, \] and let $B=\partial\mathcal M$ be its topological boundary in the parameter plane. These are sets of parameters $c$, rather than Julia sets in the dynamical $z$-plane. The remaining questions in this source item are:

1. Is $B$ locally connected? Precisely, for every $b\in B$ and every neighbourhood $U$ of $b$ in the relative topology of $B$, must there be a connected relative neighbourhood $V$ with $b\in V\subset U$?
2. Does $B$ have two-dimensional Lebesgue measure zero?

The original item also asked whether the Hausdorff dimension of $B$ is two. The source update records Shishikura's proof of that assertion, so it is supplied as established context rather than a further requested conjecture. Full Hausdorff dimension alone does not decide whether planar area vanishes. Local connectedness is asked at every boundary point, including infinitely renormalizable parameters; results for special parameter classes do not settle this unrestricted formulation.

Short English statement

Is the Mandelbrot-set boundary locally connected everywhere, and does it have zero planar area?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302088 (AMR-022-2088), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.88. Original source indexed by AMR-022-2088. https://arxiv.org/abs/1809.07200.