Dimension and ergodicity of geometrically finite 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 rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ of degree at least two, $J(f)$ is the complement of its Fatou set, where the iterates form a normal family. If $C(f)$ is its set of critical points, set $P(f)=\bigcup_{n\geq1}f^n(C(f))$. In the source convention $f$ is geometrically finite when $\overline{P(f)}\cap J(f)$ is finite; parabolic cycles are allowed.Write $\delta=\dim_H J(f)$ for Hausdorff dimension and $\mathcal H^\delta$ for Hausdorff measure of that dimension. Ergodicity on the sphere is with respect to its spherical area measure class: a measurable set invariant under inverse image is null or conull.

Statement recorded in the supplied source.

Prove that either $J(f)=\widehat{\mathbb C}$ and $f$ is ergodic there, or $\delta<2$. In the latter case determine the $\delta$-dimensional measure of $J(f)$ and describe the dynamics with respect to that measure class. [S2]

Short English statement

Establish the sphere-or-dimension-below-two dichotomy, and determine the measure and measure-class dynamics in the latter case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300084 (AMR-052-0084). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld (editor) and conference contributors, Conformal Dynamics Problem List, arXiv:math/9201271v1 (1990), Curt McMullen, §5, Geometrically finite rational maps, Problem 1, printed p. 11. https://arxiv.org/abs/math/9201271v1.