Research Problems in Function Theory — Problem 2.78

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. A periodic point $p$ of period $m$ is attracting if the multiplier of $g^m$ at $p$ has modulus less than one, computed in a local coordinate also when $p=\infty$. Its attracting basin consists of points whose forward orbits approach that periodic cycle. Let $\mathcal H_d\subset\operatorname{Rat}_d$ consist of maps for which every critical point belongs to an attracting basin. The density-of-hyperbolicity conjecture asks whether \[ \overline{\mathcal H_d}=\operatorname{Rat}_d \qquad\text{for every }d\ge2. \] Equivalently, for every degree-$d$ rational map $g$ and every neighbourhood $U$ of its coefficient class, does $U$ contain a map whose critical orbits are all attracted to periodic cycles? Perturbations retain the same degree; the assertion is not made only after restricting to real coefficients or to one prescribed one-dimensional family. Results in real polynomial subfamilies, noted in the source, do not establish density in the full complex parameter space.

Short English statement

Can every rational map of a fixed degree be approximated by rational maps whose critical points all converge to attracting cycles?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302078 (AMR-022-2078), 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 2.78. Original source indexed by AMR-022-2078. https://arxiv.org/abs/1809.07200.