Math Conjectures

Density of hyperbolicity for rational maps

OPENLandmarkConjectureProposed c. 1920 · Standard version

Canonical statement

For every d≥2d\ge2, hyperbolic maps are dense in the space Rat⁡d\operatorname{Rat}_d of degree-dd rational maps f:C^→C^f:\widehat{\mathbb C}\to\widehat{\mathbb C}, with its coefficient topology. Here ff is hyperbolic when the closure of the forward orbits of all its critical points is disjoint from its Julia set J(f)J(f).
View source LaTeX
For every \(d\ge2\), hyperbolic maps are dense in the space \(\operatorname{Rat}_d\) of degree-\(d\) rational maps \(f:\widehat{\mathbb C}\to\widehat{\mathbb C}\), with its coefficient topology. Here \(f\) is hyperbolic when the closure of the forward orbits of all its critical points is disjoint from its Julia set \(J(f)\).
DYN-003:[ManeSadSullivan1983DynamicsRationalMaps][Lyubich1983StabilityRationalMaps][McMullen1994ComplexDynamicsRenormalization][KozlovskiShenVanStrien2007DensityHyperbolicity]
Dynamical systemsAnalysis and functional analysis

Notes

The conjecture asserts that hyperbolic maps are dense in the space Rat⁡d\operatorname{Rat}_d of degree-dd rational maps of the Riemann sphere, where a map is hyperbolic when the closure of its critical orbits avoids the Julia set. The question is usually traced to Fatou's work of around 1920, with the modern density formulation crystallizing around 1980.

Hyperbolic maps form an open set on which the dynamics is structurally stable, and the theory of holomorphic motions shows that structurally stable maps are dense in any holomorphic family [ManeSadSullivan1983DynamicsRationalMaps] [Lyubich1983StabilityRationalMaps]; the conjecture therefore reduces to showing that a stable map must be hyperbolic. Renormalization and rigidity techniques have driven much of the subsequent progress [McMullen1994ComplexDynamicsRenormalization], and density of hyperbolicity is a theorem in real one-dimensional dynamics [KozlovskiShenVanStrien2007DensityHyperbolicity], as well as in various combinatorially restricted complex families.

What remains is the general parameter whose critical points exhibit recurrent dynamics, for which no hyperbolic perturbation is known to exist; the conjecture is open for every degree d≥2d\ge2.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.