Local Connectivity of the Mandelbrot Set (MLC)Open Problem
## 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.