Dimension and measure of Cremer Julia sets

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a polynomial $f$ of degree at least two, $f^n$ is its $n$-fold iterate, $K(f)=\{z\in\mathbb C:(f^n(z))_{n\geq0}\text{ is bounded}\}$ is its filled Julia set, and $J(f)=\partial K(f)$ is its Julia set. A space is locally connected if each point has a neighbourhood basis of connected sets. In Milnor's terminology here, a Cremer polynomial is $P_\alpha(z)=z^2+\lambda z$, where $\lambda=e^{2\pi i\alpha}$, $\alpha\in\mathbb R\setminus\mathbb Q$, and no holomorphic change of coordinate near $0$ conjugates $P_\alpha$ to $w\mapsto\lambda w$. Its unique critical point is $-\lambda/2$. The Hausdorff dimension $\dim_H E$ is the threshold exponent for Hausdorff measures $\mathcal H^s(E)$, defined using covers by sets of small diameter and sums of their $s$th powers. Planar Lebesgue measure is denoted by $m_2$.

Statement recorded in the supplied source.

Does every Cremer polynomial $P_\alpha$ satisfy $\dim_HJ(P_\alpha)=2$? Does every such Julia set satisfy $m_2(J(P_\alpha))=0$? [S2]

Short English statement

Must every Cremer quadratic Julia set have full planar Hausdorff dimension, and must every one nevertheless have area zero?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300032 (AMR-052-0032). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John Milnor, "Problems on local connectivity", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 7, pp. 21; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.