Density of hyperbolic rational maps

Definitions and mathematical statement

For $d\ge2$, let $\operatorname{Rat}_d$ be the degree-$d$ rational self-maps of the Riemann sphere. Represent them by pairs of degree-$d$ homogeneous polynomials with no common zero, modulo simultaneous nonzero scalar multiplication, and use the resulting open-subset topology in $\mathbb P^{2d+1}(\mathbb C)$. Let $\operatorname{Poly}_d$ be the polynomials $a_dz^d+\cdots+a_0$ with $a_d\ne0$, with their coefficient topology.

A rational map $f$ is hyperbolic here if every critical point on the sphere has forward orbit converging to an attracting periodic cycle. A cycle of period $k$ is attracting when the modulus of its multiplier $(f^k)'$ is less than one, computed in local coordinates at a point of the cycle.

Prove that hyperbolic maps are dense in each of $\operatorname{Rat}_d$ and $\operatorname{Poly}_d$, for every $d\ge2$. Explicitly, for every map $f$ in either space and every neighborhood $U$ of $f$ within that same space, there must exist a hyperbolic $g\in U$ of the same degree.

In the polynomial case, infinity is a superattracting fixed point; finite critical points escaping to infinity satisfy the attraction condition. Approximation of a polynomial must be by polynomials, not merely by rational maps. No restriction to real coefficients, connected Julia sets, or any selected hyperbolic component is intended.

Short English statement

Can every complex rational map, and every complex polynomial, be approximated within its degree and class by hyperbolic maps?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300083 (AMR-052-0083), 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, 1990 §5 conjecture. Original source indexed by AMR-052-0083. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 McMullen, Section5, density conjecture, in Conformal Dynamics Problem List, IMS90/1, p.11. https://www.math.stonybrook.edu/preprints/ims90-1.pdf.