Diameter of Mandelbrot limbs

Definitions and mathematical statement

For $P_c(z)=z^2+c$, let $M=\{c\in\mathbb C:(P_c^n(0))_{n\ge0}\text{ is bounded}\}$ be the Mandelbrot set. Its principal cardioid parameterizes attracting fixed points and has boundary $c(\mu)=\mu/2-\mu^2/4$ for $|\mu|=1$. For relatively prime integers $0<p<q$, put \[ c_{p/q}=\tfrac12e^{2\pi ip/q}-\tfrac14e^{4\pi ip/q}. \] The closed $p/q$-limb $M(p/q)$ is the part of $M$ attached to the principal cardioid at $c_{p/q}$ on the side away from that cardioid, including its attachment point. Equivalently it is the intersection of $M$ with the closed parameter wake bounded by the two characteristic external parameter rays landing at $c_{p/q}$.

Does there exist an absolute constant $K>0$ such that \[ \operatorname{diam}M(p/q)\le K/q^2 \quad\text{for every reduced }0<p/q<1? \] If this stronger estimate fails, does the weaker uniform estimate $\operatorname{diam}M(p/q)\le K\log(q)/q^2$ hold? Diameter is Euclidean distance in the parameter plane, not angular width or hyperbolic diameter. The constant must be independent of both numerator and denominator. The limb contains all decorations in its wake, not merely the period-$q$ hyperbolic component attached to the cardioid. Bounds for the visible bulb alone therefore do not give the stated estimate for the entire limb. The denominator is taken after reducing the rational angle, so artificial repetitions of a fraction cannot improve the proposed bound.

Short English statement

Do all principal Mandelbrot limbs have Euclidean diameter $O(q^{-2})$, or at least $O(q^{-2}\log q)$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300040 (AMR-052-0040), 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, 1992, Milnor local P15. Original source indexed by AMR-052-0040. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Milnor, Problems on local connectivity, Problem15, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.