Math Conjectures

Mandelbrot local-connectivity conjecture

OPENLandmarkConjectureProposed c. 1982 · Full conjecture

Canonical statement

The Mandelbrot set
M={c∈C:{fcn(0):n≥0} is bounded},fc(z)=z2+c, \mathcal M=\{c\in\mathbb C: \{f_c^n(0):n\ge0\}\text{ is bounded}\},\qquad f_c(z)=z^2+c,
is locally connected: for every c∈Mc\in\mathcal M and every neighborhood UU of cc, there is a connected neighborhood VV of cc in the subspace M\mathcal M with V⊂UV\subset U.
View source LaTeX
The Mandelbrot set \[ \mathcal M=\{c\in\mathbb C: \{f_c^n(0):n\ge0\}\text{ is bounded}\},\qquad f_c(z)=z^2+c, \] is locally connected: for every \(c\in\mathcal M\) and every neighborhood \(U\) of \(c\), there is a connected neighborhood \(V\) of \(c\) in the subspace \(\mathcal M\) with \(V\subset U\).
DYN-004:[DouadyHubbard1985EtudeMandelbrot][Yoccoz1995PetitsDiviseurs][Lyubich1997DynamicsQuadratics][DudkoLyubich2023MLC]
Dynamical systemsAnalysis and functional analysis

Notes

The conjecture, usually abbreviated MLC, asserts that the Mandelbrot set M\mathcal M — the set of parameters cc for which the critical orbit of fc(z)=z2+cf_c(z)=z^2+c stays bounded — is locally connected. It was formulated in the early 1980s during Douady and Hubbard's foundational study of M\mathcal M [DouadyHubbard1985EtudeMandelbrot], where local connectivity was shown to yield a complete combinatorial description of the set.

Large parts of the conjecture are known. Yoccoz proved local connectivity at every parameter that is at most finitely renormalizable [Yoccoz1995PetitsDiviseurs]; Lyubich's work on quadratic dynamics brought many infinitely renormalizable combinatorics under control [Lyubich1997DynamicsQuadratics]; and local connectivity has more recently been established at Feigenbaum points [DudkoLyubich2023MLC]. Local connectivity also holds at hyperbolic parameters.

What resists is the residual class of infinitely renormalizable parameters whose combinatorics do not yet admit uniform a-priori bounds on renormalization. The full conjecture is open, and resolving it amounts to supplying such bounds across all remaining combinatorial types.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.