Explicit full-dimensional Julia set

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 rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ of degree at least two, $J(f)$ is the complement of the largest open set on which the iterates form a normal family. Use the spherical metric for Hausdorff dimension and the spherical area measure for Lebesgue measure. 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.

Find an explicit rational map $f$ with $\dim_HJ(f)=2$. When a Julia set has Hausdorff dimension two but Lebesgue measure zero, what is a natural geometric measure on it? [S2]

Short English statement

Give an explicit rational Julia set of dimension two, and identify a natural geometric measure for such sets when their ordinary area is zero.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300045 (AMR-052-0045). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mikhail Lyubich, "Measure and Dimension of Julia Sets", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 5, pp. 27; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.