Local connectivity of quadratic Julia sets

Definitions and mathematical statement

For $c\in\mathbb C$, define $P_c(z)=z^2+c$, its filled Julia set $K_c=\{z:\sup_{n\ge0}|P_c^n(z)|<\infty\}$, and its Julia set $J_c=\partial K_c$. Restrict to parameters for which $J_c$ is connected, equivalently $c$ belongs to the Mandelbrot set $M=\{c:(P_c^n(0))_{n\ge0}\text{ is bounded}\}$. A topological space is locally connected if every point has a neighborhood basis consisting of connected open sets in the relative topology.

The problem is to characterize the locus \[ \mathcal L=\{c\in M:J_c\text{ is locally connected}\}. \] Seek necessary and sufficient dynamical, combinatorial, or arithmetic conditions on $c$ or its critical orbit, rather than simply restating the topological definition. Such conditions must cover all connected quadratic Julia sets, including infinitely renormalizable cases and irrationally indifferent periodic points.

The conclusion is a property of the dynamical-plane Julia set $J_c$, not local connectivity of the Mandelbrot set at the parameter $c$. Connectedness alone is insufficient. If local connectivity holds, the external Böttcher coordinate extends continuously to the circle and gives a useful boundary model; this is one motivation for the classification. Known sufficient criteria and known non-locally-connected examples delimit parts of the locus, but the source's question asks for a complete distinction. A criterion only for finitely renormalizable maps would leave the general problem unresolved.

Short English statement

Characterize exactly which connected quadratic Julia sets are locally connected.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300072 (AMR-052-0072), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1990 §1 Q8. Original source indexed by AMR-052-0072. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Bielefeld, Conformal Dynamics Problem List (1990), Section1, Question8. https://web.pdx.edu/~veerman/bielefeld.pdf.