Open problem, Differential equations and dynamical systems, Analysis, posed 1982

Local connectivity of the Mandelbrot set (MLC)

Open

Let $M = \{c \in \mathbb{C} : \sup_{n \ge 1} |f_c^{\circ n}(0)| < \infty\}$ be the Mandelbrot set for the quadratic family $f_c(z) = z^2 + c$ . Is $M$ locally connected at every point $c \in M$ (meaning that every point $c \in M$ has arbitrarily small connected neighborhoods in the subspace topology of $M \subset \mathbb{C}$ )?

Research frontier as of 2026

As of 2026, the MLC conjecture remains open in full generality. Yoccoz's theorem settles all finitely renormalizable parameters $c \in M$ , reducing MLC to infinitely renormalizable parameters (points nested inside an infinite sequence of small copies of $M$ ). While a priori complex bounds and local connectivity have been established for many infinitely renormalizable regimes—including bounded-type, high-type primitive, and various satellite combinatorics (via work of Lyubich, Kahn, Avila, Dudko, and others)—the general unbounded satellite infinitely renormalizable case remains the primary obstacle.

Best known results

Yoccoz (1990): the Mandelbrot set $M$ is locally connected at every parameter $c \in M$ that is at most finitely renormalizable.
Lyubich (1997) and Kahn–Lyubich (2006–2009): MLC holds at infinitely renormalizable parameters of bounded combinatorial type and at primitive infinitely renormalizable parameters satisfying decoration bounds.

Tools and where they stop

Tool	Achieved	Where it stops
Yoccoz puzzle and parapuzzle with Grötzsch ring moduli sums	Proves that nested parapuzzle pieces shrink to a single point for all finitely renormalizable parameters	For infinitely renormalizable parameters, the annuli between consecutive puzzle depths can degenerate unless a priori geometric bounds hold
Quadratic-like renormalization and the Kahn–Lyubich Covering Lemma	Establishes a priori complex bounds and rigidity for primitive and bounded-satellite renormalizations	Unbounded cascades of satellite renormalizations can distort conformal moduli faster than standard covering lemmas control

Open questions

Do a priori complex bounds hold for arbitrary infinitely renormalizable quadratic polynomials with unbounded satellite combinatorics?
Are hyperbolic parameters dense in the Mandelbrot set $M$ (the Density of Hyperbolicity Conjecture in degree $2$ )?

References

Adrien Douady, John H. Hubbard (1984). Étude dynamique des polynômes complexes (Première et deuxième parties)
John H. Hubbard (1993). Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz
Mikhail Lyubich (1997). Dynamics of quadratic polynomials, I–II · DOI:10.1007/BF02392711