Lebesgue ergodicity on a spherical Julia set

Definitions and mathematical statement

Let $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational map of degree $d\ge2$, and assume its Julia set is the whole Riemann sphere. Equivalently there is no nonempty open set on which the iterates form a normal family. Let $m$ be normalized spherical area measure. Although $R$ need not preserve $m$, it is nonsingular: inverse images of area-zero sets have area zero.

The first question is whether $R$ is ergodic for this measure class, meaning that every measurable $E$ with $m(R^{-1}E\mathbin{\triangle} E)=0$ has $m(E)=0$ or $m(E)=1$. This definition does not assert that spherical area itself is an invariant probability measure.

If universal ergodicity fails, the source asks the weaker bound of at most $2d-2$ ergodic components. Precisely, can the sphere, modulo an area-null set, be partitioned into $r\le2d-2$ invariant measurable sets of positive area, on each of which the restricted nonsingular action is ergodic? The parts are understood modulo null sets, and invariance is under full inverse image in that same sense.

The hypothesis excludes attracting basins, Siegel disks, and all other Fatou components. Ergodicity for a different preferred invariant measure, such as the measure of maximal entropy, does not answer the area-class question. The first assertion is stronger than the finite-component alternative, and either requires control of all measurable invariant subsets, not only closed invariant sets.

Short English statement

Must a rational map with spherical Julia set be ergodic for the area measure class, or have at most $2d-2$ components?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300043 (AMR-052-0043), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1992, Lyubich P3. Original source indexed by AMR-052-0043. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Lyubich, Problems on measurable dynamics, Problem3, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.