Invariant line fields on Julia sets

Definitions and mathematical statement

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational map of degree at least two. Its Julia set $J(f)$ is the complement of the open set on which the iterates form a normal family. A measurable invariant line field on $J(f)$ means a measurable choice of an unoriented real tangent line $\ell(z)\subset T_z\widehat{\mathbb C}$ on an invariant measurable subset $E\subset J(f)$ of positive spherical area, such that \[ Df_z(\ell(z))=\ell(f(z)) \quad\text{for almost every }z\in E. \] Invariance of $E$ is modulo area-zero sets. Critical points, a finite set, do not affect the almost-everywhere condition.

A Lattès map is a rational map arising from an affine expanding map $L(w)=aw+b$ of a complex torus $\mathbb C/\Lambda$, where $a\Lambda\subset\Lambda$ and $|a|>1$, by a finite holomorphic quotient $\pi:\mathbb C/\Lambda\to\widehat{\mathbb C}$ satisfying $f\pi=\pi L$.

Does the existence of such a line field force $f$ to be a Lattès map? Equivalently, prove that every rational map outside this exceptional class has no measurable invariant line field supported on a positive-area part of its Julia set. The positive-area requirement excludes fields specified merely on countable or zero-area invariant sets. No continuity of the field is assumed, and line fields on Fatou components are outside the question.

Short English statement

Are rational maps with a measurable invariant line field on a positive-area part of their Julia set necessarily Lattès maps?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300044 (AMR-052-0044), 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 P4. Original source indexed by AMR-052-0044. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Lyubich, Measure and Dimension of Julia Sets, Problem4 (Sullivan), in Problems in Holomorphic Dynamics, IMS92/7, p.27. https://www.math.stonybrook.edu/preprints/ims92-7.pdf.