Research Problems in Function Theory — Problem 2.80

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 $J(g)=\widehat{\mathbb C}$. A bounded measurable Beltrami differential is locally written $\mu(z)\,d\overline z/dz$; its coefficients transform according to this tensor expression between charts. Define the complex vector space \[ \mathcal B(g)=\left\{\mu\in L^\infty: \mu(z)=\mu(g(z))\frac{\overline{g'(z)}}{g'(z)} \text{ for almost every noncritical }z\right\}. \] Differentials equal almost everywhere are identified. No normalization $\|\mu\|_\infty<1$ is imposed when discussing vector-space dimension; such a normalization would define a ball rather than a vector space. Must $\dim_{\mathbb C}\mathcal B(g)\le1$, and must a nonzero space occur only for the appropriate Lattès maps? A Lattès map is obtained from an expanding affine holomorphic self-map of a complex torus by a finite holomorphic quotient from the torus to the sphere. The question asks for the precise exceptional members of this class as well as exclusion of all other rational maps. It is an invariant-differential question with respect to spherical area measure.

Short English statement

When the Julia set is the whole sphere, is there at most one independent invariant Beltrami differential, with nonzero examples confined to Lattès maps?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302080 (AMR-022-2080), 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 2.80. Original source indexed by AMR-022-2080. https://arxiv.org/abs/1809.07200.