MLC Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $c\in\mathbb C$, define $z_0(c)=0$ and $z_{m+1}(c)=z_m(c)^2+c$ for integers $m\ge0$. The Mandelbrot set is \[ \mathcal M=\{c\in\mathbb C:\sup_{m\ge0}|z_m(c)|<\infty\}, \] with the topology inherited from $\mathbb C$. A neighborhood below means a relatively open set in $\mathcal M$.

Statement recorded in the supplied source.

Is $\mathcal M$ locally connected? Explicitly, for every $c\in\mathcal M$ and every relatively open neighborhood $U$ of $c$ in $\mathcal M$, does there exist a connected relatively open set $V$ such that $c\in V\subseteq U$? [S2]

Short English statement

Does every point of the Mandelbrot set have arbitrarily small connected neighborhoods within the Mandelbrot set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1338 (DYN-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dierk Schleicher, On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets (1999), arXiv:math/9902155v1, Sections 1--2. https://arxiv.org/abs/math/9902155v1.