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OpenAnalysis·1982

Mandelbrot Local Connectivity

Asks whether the Mandelbrot set is locally connected at every point.

Formula

Mandelbrot set
M={c∈C:sup⁡n∣zn∣<∞,  z0=0,  zn+1=zn2+c}M=\{c\in\mathbb{C}: \sup_n|z_n|<\infty,\; z_0=0,\; z_{n+1}=z_n^2+c\}

The Mandelbrot set is the set of parameters c for which the critical orbit of z² + c remains bounded.

MLC conjecture
M is locally connectedM\text{ is locally connected}

If true, M has no infinitely fine filaments and the Douady–Hubbard landing theorem extends to all external rays.

Density of hyperbolicity (implication)
MLC  ⟹  {c:z2+c is hyperbolic} is dense in ∂M\text{MLC}\;\Longrightarrow\;\{c: z^2+c\text{ is hyperbolic}\}\text{ is dense in }\partial M

MLC would imply that hyperbolic dynamics is dense in the quadratic family — a central conjecture in holomorphic dynamics.

Summary

The MLC conjecture asks for a precise topological regularity property of the Mandelbrot set. The image is iconic, but the conjecture is not merely visual: local connectivity would organize how parameter space is navigated by external rays.

Sources

Exploring the Mandelbrot set

1985

Etude dynamique des polynomes complexes (Orsay Notes)

1984

Dynamics of quadratic polynomials, I-II

1999

Videos

Thumbnail for Beyond the Mandelbrot set, an intro to holomorphic dynamics

Beyond the Mandelbrot set, an intro to holomorphic dynamics

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Thumbnail for What's so special about the Mandelbrot Set? - Numberphile

What's so special about the Mandelbrot Set? - Numberphile

Numberphile
Thumbnail for The Mandelbrot Set - Numberphile

The Mandelbrot Set - Numberphile

Numberphile