Research Problems in Function Theory — Problem 2.79

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.

Let $g:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational map of degree at least $2$. Its Julia set $J(g)$ is the complement of the normality set of its iterates. Area means spherical two-dimensional area. A Beltrami differential on $J(g)$ is a measurable coefficient $\mu$ with $\|\mu\|_\infty<1$. It is invariant if, in local complex coordinates, \[\mu(z)=\mu(g(z))\frac{\overline{g'(z)}}{g'(z)}\] for almost every $z\in J(g)$; the finitely many critical points do not affect this condition. The source update reports Buff--Chéritat quadratic counterexamples to part (b), with positive-area Julia sets.

Statement recorded in the supplied source.

Assume $J(g)\ne\widehat{\mathbb C}$. (a) Must every invariant Beltrami differential on $J(g)$ vanish almost everywhere? (b) Must $J(g)$ have area zero? [S2]

Short English statement

For a rational map whose Julia set is not the whole sphere, must every invariant Beltrami differential vanish, and must the Julia set have zero area?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302079 (AMR-022-2079). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.79 and Update 2.79. https://arxiv.org/abs/1809.07200.