No invariant line fields, density of hyperbolicity, and the geometry of the dynamical moduli space: the open core of the McMullen–Sullivan Teichmüller theory of rational maps

McMullen and Sullivan (Adv. Math. 135 (1998) 351–395) endowed each rational map $f$ with a dynamical Teichmüller space $\operatorname{Teich}(\hat{\mathbb{C}},f)$ and modular group $\operatorname{Mod}(\hat{\mathbb{C}},f)$ , proved that the quotient injects holomorphically into the moduli space $\operatorname{Rat}_d/\operatorname{Aut}(\hat{\mathbb{C}})$ , and showed that its dimension counts critical moduli in the Fatou set plus independent measurable invariant line fields on the Julia set. Several problems posed in that paper are now settled: the injection is an immersion (Makienko 2010; Astorg 2017), the image can accumulate on itself (Branner's cubic examples, recorded in the paper's own footnote 1), and McMullen's related 1987 conjecture that the dynamical moduli space is Carathéodory hyperbolic (i.e. bounded holomorphic functions separate its points) was proved by Ji–Xie (2024/2026). The surviving open core has four linked components: (1) the no invariant line fields conjecture — a rational map carries no measurable invariant line field on its Julia set unless it is a Lattès map, i.e. double covered by an integral torus endomorphism; (2) Fatou's density of hyperbolicity conjecture, implied by (1); (3) finiteness of the modular group when the Julia set is the whole sphere, also implied by (1); (4) characterization of when the moduli image is locally closed. All four components remain open, with substantial partial results in the real one-dimensional case and for special classes of maps.

Background

A rational map is a holomorphic self-map $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ of the Riemann sphere given by a ratio of two coprime polynomials, of degree $d\ge 2$ . Under iteration the sphere splits into two completely invariant sets: the Fatou set is the largest open set on which the family of iterates $(f^n)$ is normal (every sequence of iterates has a locally uniformly convergent subsequence), and on it the dynamics is eventually simple — each component is the basin of an attracting, superattracting or parabolic cycle (according as the multiplier of the cycle has modulus $<1$ , equals $0$ so that a critical point lies on the cycle, or is a root of unity), or carries an irrational rotation — a Siegel disk (a disk on which an iterate of $f$ is conformally conjugate to a rotation by an irrational angle) or a Herman ring (an annulus with the same property); the Julia set $J(f)$ is its complement, a perfect, typically fractal set on which the dynamics is chaotic. A rational map is hyperbolic (also called expanding or Axiom A) if it is uniformly expanding on its Julia set; equivalently, all critical points of $f$ lie in the Fatou set and converge to attracting or superattracting cycles. Hyperbolic maps are structurally stable (any map sufficiently close to $f$ in $\operatorname{Rat}_d$ is topologically conjugate to $f$ ) and form an open subset of $\operatorname{Rat}_d$ , the complex manifold of all degree- $d$ rational maps; the group $\operatorname{Aut}(\hat{\mathbb{C}})$ of Möbius transformations acts on $\operatorname{Rat}_d$ by conjugation, and the quotient $\operatorname{Rat}_d/\operatorname{Aut}(\hat{\mathbb{C}})$ is the moduli space of conformal conjugacy classes. Since Fatou's work in the 1920s the central structural question of the field has been whether hyperbolic maps are dense in $\operatorname{Rat}_d$ for every degree — Fatou's density of hyperbolicity conjecture.

The quasiconformal reformulation is the heart of the matter. A quasiconformal homeomorphism is a homeomorphism of the sphere with locally integrable distributional derivatives and bounded dilatation $\|\bar\partial\phi/\partial\phi\|_\infty<\infty$ ; by the measurable Riemann mapping theorem every measurable Beltrami coefficient $\mu$ (i.e. $\mu(z)\,d\bar z/dz$ with $\|\mu\|_\infty<1$ ) is realized as $\bar\partial\phi/\partial\phi$ of a quasiconformal $\phi$ . A Beltrami differential invariant under $f$ therefore deforms the complex structure along the dynamics and produces maps quasiconformally conjugate to $f$ . A measurable invariant line field on the Julia set is such an invariant coefficient with $|\mu|=1$ on a subset of $J(f)$ of positive planar measure and $\mu=0$ off $J(f)$ : a field of tangent directions on the Julia set preserved by the dynamics, invisible to the holomorphic category. The only known source of such line fields is arithmetic: if $X=\mathbb{C}/\Lambda$ is a complex torus, $F(z)=az+b$ is an affine self-map of $X$ with integral linear part (an integral torus endomorphism), and $\pi:X\to\hat{\mathbb{C}}$ is the quotient by the finite group generated by $z\mapsto -z$ and translations, then $f=\pi\circ F\circ\pi^{-1}$ is a rational map — a Lattès map, said by McMullen and Sullivan to be double covered by an integral torus endomorphism (in the later literature these exceptional maps are also called flexible Lattès maps) — with $J(f)=\hat{\mathbb{C}}$ , postcritically finite dynamics (every critical point of $f$ has a finite, i.e. eventually periodic, forward orbit), and the line field descended from $d\bar z/dz$ on the torus invariant under $f$ .

McMullen and Sullivan built a Teichmüller theory for these dynamical systems, parallel to the classical theory of Riemann surfaces and to Sullivan's deformation theory of Kleinian groups (discrete groups of Möbius transformations acting as isometries of hyperbolic 3-space). They define $\operatorname{Teich}(\hat{\mathbb{C}},f)$ as the unit ball of $f$ -invariant Beltrami differentials modulo the group $\operatorname{QC}_0(\hat{\mathbb{C}},f)$ of quasiconformal automorphisms commuting with $f$ that are isotopic to the identity through such automorphisms, and the modular group $\operatorname{Mod}(\hat{\mathbb{C}},f)=\operatorname{QC}(\hat{\mathbb{C}},f)/\operatorname{QC}_0(\hat{\mathbb{C}},f)$ of isotopy classes of quasiconformal self-conjugacies. Their structure theorem identifies $\operatorname{Teich}(\hat{\mathbb{C}},f)$ with a finite-dimensional complex manifold: the product of a polydisk (a product of complex disks, recording the moduli of dynamically foliated annuli coming from Siegel disks, Herman rings and superattracting basins, and the invariant line fields on the Julia set) with the classical Teichmüller spaces of finitely many punctured spheres and tori describing the attracting and parabolic basins. The modular group acts properly discontinuously by holomorphic automorphisms, and the quotient orbifold injects holomorphically into $\operatorname{Rat}_d/\operatorname{Aut}(\hat{\mathbb{C}})$ , its image being exactly the quasiconformal conjugacy class of $f$ ; in particular $\dim\operatorname{Teich}(\hat{\mathbb{C}},f)=n_{AC}+n_{LF}$ , the number of grand orbits of acyclic critical points in the Fatou set (two points lie in the same grand orbit when some iterates of them coincide; a critical point is acyclic when its forward orbit is infinite) plus the number of independent invariant line fields on the Julia set. Combined with the $J$ -stability theory of Mañé–Sad–Sullivan — in any holomorphic family of rational maps, the parameters whose maps are quasiconformally conjugate on their Julia sets to a fixed nearby map form an open dense set, whose complement is the bifurcation locus — a dimension count shows that if the only invariant line fields were the Lattès ones, then every $J$ -stable map would have all critical points in attracting basins, hence be hyperbolic, so hyperbolicity would be dense (Theorem 9.3 of the paper). For finitely generated Kleinian groups the analogous no-invariant-line-fields statement is a theorem of Sullivan; for rational maps it has remained the central open rigidity problem of one-dimensional complex dynamics, alongside Fatou's density conjecture itself.

Problem Statement

Resolve the open core of the problem list left by the McMullen–Sullivan Teichmüller theory of rational maps — the following four linked questions, each to be proved, refuted, or (for the fourth) characterized with proof:

(1) No invariant line fields conjecture. Prove or refute: a rational map $f$ of degree $d\ge 2$ carries no measurable invariant line field on its Julia set $J(f)$ — i.e. no Beltrami coefficient $\mu$ with $|\mu|=1$ on a positive-planar-measure subset of $J(f)$ , $\mu=0$ off $J(f)$ , and $f^*\mu=\mu$ almost everywhere — unless $f$ is double covered by an integral torus endomorphism (a Lattès map).

(2) Fatou's density of hyperbolicity conjecture. Prove or refute: for every $d\ge 2$ , the hyperbolic (expanding) rational maps are dense in $\operatorname{Rat}_d$ . By McMullen–Sullivan (Theorem 9.3), a positive answer to (1) implies a positive answer to (2); for the quadratic family the two conjectures are equivalent.

(3) Finiteness of the modular group. Prove or refute: if $J(f)=\hat{\mathbb{C}}$ and $f$ is not double covered by an integral torus endomorphism, then $\operatorname{Mod}(\hat{\mathbb{C}},f)$ is finite. (For the torus-endomorphism maps the modular group contains $\mathrm{PSL}_2(\mathbb{Z})$ — the classical modular group of integer $2\times 2$ matrices of determinant one, taken modulo sign — with finite index; a positive answer to (1) implies this one, since then $\operatorname{Mod}(\hat{\mathbb{C}},f)=\operatorname{Aut}(f)$ , the group of Möbius transformations commuting with $f$ .)

(4) Local closedness of the moduli image. Characterize the rational maps $f$ for which the image of the injection $\operatorname{Teich}(\hat{\mathbb{C}},f)/\operatorname{Mod}(\hat{\mathbb{C}},f)\to\operatorname{Rat}_d/\operatorname{Aut}(\hat{\mathbb{C}})$ is locally closed (i.e. the intersection of an open set with a closed set, equivalently closed within some open neighborhood) in $\operatorname{Rat}_d/\operatorname{Aut}(\hat{\mathbb{C}})$ ; by Branner's cubic examples it is not locally closed in general.

The common scientific objective is to complete the rigidity and deformation theory of rational maps: determine all quasiconformal deformations of a rational map — showing that none come from the Julia set except for the Lattès class (items 1–3, which are linked by the implications above) — and describe the global complex geometry of the resulting dynamical moduli space inside the space of all rational maps (item 4). These four questions, taken at full generality — no restriction on the degree, the measure class of the Julia set, or the model class beyond the stated Lattès exception — carry the following logical relations: an affirmative resolution of (1) discharges (2) and (3) through the implications stated above, a refutation of (1) leaves (2) and (3) to be settled independently, and (4) stands alone.

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