Research Problems in Function Theory — Problem 2.81

Definitions and mathematical statement

For an integer $d\ge2$, let $\operatorname{Rat}_d$ be the space of degree-$d$ rational maps of the Riemann sphere. Represent a map by coprime homogeneous degree-$d$ polynomials, modulo common nonzero scaling; the induced coefficient topology is used. A critical point has local mapping degree greater than one. The Julia set $J(g)$ is the complement of the largest open set on which the iterates $\{g^n:n\ge0\}$ form a normal family in the spherical metric. Assume $J(g)=\widehat{\mathbb C}$. Let $m$ denote spherical area measure, normalized if desired to have total mass one. Does every Borel set $B\subset\widehat{\mathbb C}$ satisfying \[ g^{-1}(B)=B \] have either $m(B)=0$ or $m(\widehat{\mathbb C}\setminus B)=0$? This is ergodicity of the map for the spherical-area measure class. The measure itself need not be preserved by $g$, so the question is not restricted to area-preserving maps or to invariance of an integral. Since a nonconstant rational map is nonsingular for area, invariant sets modulo null sets give the corresponding measure-class formulation. The hypothesis that the Julia set fills the sphere removes the open Fatou components, but does not by itself rule out a measurable invariant partition with both pieces of positive area. The requested conclusion concerns every invariant Borel set. Ergodicity of a different canonical invariant probability measure, such as the measure of maximal entropy, is a separate statement and is not the question here.

Short English statement

Is a rational map with Julia set equal to the whole sphere ergodic for the spherical-area measure class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302081 (AMR-022-2081), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.81. Original source indexed by AMR-022-2081. https://arxiv.org/abs/1809.07200.