Research Problems in Function Theory — Problem 2.82

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. A point $c$ is preperiodic for $g$ if its forward orbit $\{c,g(c),g^2(c),\ldots\}$ is finite. It is periodic if $g^m(c)=c$ for some $m\ge1$. Let \[ \mathcal L_d=\{g\in\operatorname{Rat}_d: \text{every critical point is preperiodic and none is periodic}\}. \] Multiplicity of a critical point does not change the orbit condition; the critical point at infinity is included when present. Herman's approximation question asks whether \[ J(f)=\widehat{\mathbb C}\quad\Longrightarrow\quad f\in\overline{\mathcal L_d} \qquad(f\in\operatorname{Rat}_d). \] Thus, given any such $f$ and any neighbourhood in the fixed-degree coefficient space, one seeks a nearby rational map whose critical orbits are all finite but avoid periodic critical points. The approximants themselves need not have the same critical-point multiplicities as $f$. The closure is taken among degree-$d$ maps, rather than in a compactification allowing common factors and a drop of degree. The assertion is stronger than merely finding maps with many, but not all, preperiodic critical points.

Short English statement

Can every rational map whose Julia set is the whole sphere be approximated by maps with strictly preperiodic critical points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302082 (AMR-022-2082), 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 2.82. Original source indexed by AMR-022-2082. https://arxiv.org/abs/1809.07200.