The set

For a complex parameter $c$, iterate the quadratic map

$$f_c(z) = z^2 + c$$

starting at the critical point $z = 0$. The Mandelbrot set is the set of parameters for which this orbit stays bounded:

$$M = \{\, c \in \mathbb{C} \ : \ \sup_{k \in \mathbb{N}} \left| f_c^{\,k}(0) \right| < \infty \,\}.$$

Equivalently -- and this is the first milestone of the mission -- $c \in M$ if and only if $|f_c^{\,k}(0)| \le 2$ for every $k$, which exhibits $M$ as a compact subset of the plane.

$M$ is the parameter-space picture of the simplest non-trivial family in complex dynamics, and it acts as a dictionary: the shape of $M$ near a parameter $c$ encodes the dynamics of $f_c$ on its Julia set, so structural questions about $M$ are questions about the whole quadratic family at once. Douady and Hubbard proved in 1982 that $M$ is connected, by exhibiting a conformal isomorphism

$$\Phi : \mathbb{C} \setminus M \longrightarrow \mathbb{C} \setminus \overline{\mathbb{D}}$$

between the complement of $M$ and the exterior of the closed unit disk.

The question

MLC conjecture. $M$ is locally connected: every point of $M$ has a neighbourhood basis, in the subspace topology, consisting of connected sets.

By Caratheodory's theorem, MLC is equivalent to the statement that $\Phi^{-1}$ extends continuously to the unit circle. That extension would deliver a complete combinatorial description of $M$ -- the pinched disk model of Douady and Thurston -- in which every boundary point is labelled by the external rays landing on it. Two headline consequences follow: the density of hyperbolicity in the quadratic family (Fatou's conjecture: every quadratic polynomial can be perturbed to one with an attracting cycle), and zero area for $\partial M$.

MLC has been open since the early 1980s and is regarded as the central problem of one-dimensional complex dynamics.

Timeline

• 1982 -- Douady and Hubbard prove that $M$ is connected, via the Boettcher uniformisation of its complement, and formulate MLC.
• 1984/85 -- The Orsay notes develop the combinatorics of external rays and the pinched-disk model, and show that MLC implies the density of hyperbolicity in the quadratic family.
• 1990 -- Yoccoz proves MLC at every finitely renormalizable parameter without an indifferent periodic point, introducing the Yoccoz puzzle and the rigidity techniques that dominate later work.
• 1997 -- Lyubich extends local connectivity to infinitely renormalizable parameters of bounded type, using complex bounds for quadratic-like renormalization.
• 1997 -- Graczyk-Swiatek and Lyubich prove density of hyperbolicity in the real quadratic family.
• 1998 -- Shishikura proves that $\partial M$ has Hausdorff dimension $2$, by parabolic implosion. Whether $\partial M$ has positive area remains open.
• 2005 -- Buff and Cheritat construct quadratic Julia sets of positive area, showing that the analogous area question in the dynamical plane has a negative answer.
• Today -- MLC is known at large classes of parameters, but the general case, and with it the density of hyperbolicity, remain open.

What this mission asks for

The goal theorem is MLC itself, in the form "the Mandelbrot set, as a topological subspace of $\mathbb{C}$, is a locally connected space".

The milestones are of three kinds, and are ordered accordingly:

1. Foundations provable today -- the escape criterion (in the quadratic and the general unicritical degree) and compactness. These make the filter-theoretic definition usable and are the natural entry point for a solver new to the mission.
2. Known theorems from the literature -- connectedness of $M$ (Douady-Hubbard), the implication MLC $\Rightarrow$ density of hyperbolicity (Douady-Hubbard), and $\dim_H(\partial M) = 2$ (Shishikura). These are hard but settled, and formalizing them builds the infrastructure -- Boettcher coordinates, external rays, parabolic implosion -- that any attack on the goal will need.
3. The open companions -- density of hyperbolicity in the quadratic and unicritical families, zero area of $\partial M$, and MLC for all Multibrot sets $M_n$, the parameter sets of $z \mapsto z^n + c$.

All statements are phrased against a single shared definition file, so a solver can move between milestones without re-fixing conventions.