Is $\sup_{f\in\mathcal{H}}\dim_H J(f)=2$ for every hyperbolic component $\mathcal{H}$ of the space of rational maps?

The Hausdorff dimension of the Julia set of a hyperbolic rational map is real-analytic as the map moves inside a hyperbolic component of the moduli space of degree-$d$ rational maps, always stays strictly below $2$, and — by Theorem 6.2 of Qiu and Yang (Trans. Amer. Math. Soc. 374 (2021) 5191–5223) — has supremum exactly $2$ over every component in which each map has a simply connected periodic Fatou component whose closure is disjoint from the other Fatou components, a class covering all Cantor circle and Sierpiński carpet components. Qiu and Yang state the belief that the same supremum equality holds for every hyperbolic component of $\operatorname{Rat}_d$, for every degree $d\ge2$. This problem asks exactly that question: is $\sup_{f\in\mathcal{H}}\dim_H J(f)=2$ for every hyperbolic component $\mathcal{H}$, in particular for components whose periodic Fatou components are all multiply connected, such as Herman ring components or components with only infinitely connected attracting basins?

Background

A rational map is a holomorphic self-map $f$ of the Riemann sphere $\widehat{\mathbb{C}}$ given by a ratio of two polynomials, with topological degree $d\ge2$. Iterating $f$ splits the sphere into two completely invariant sets: the Fatou set, where the family of iterates is normal (equicontinuous in the spherical metric), and its complement, the Julia set $J(f)$, the locus of chaotic behavior, which is typically a fractal. Its fractal size is measured by the Hausdorff dimension $\dim_H$, which for subsets of the sphere takes values in $[0,2]$.

Write $\operatorname{Rat}_d$ for the space of all rational maps of degree $d$ and $\mathcal{M}_d=\operatorname{Rat}_d/\mathrm{PSL}_2(\mathbb{C})$ for its moduli space, which identifies maps differing by conjugation with a Möbius transformation. A rational map is hyperbolic if every critical point is attracted to an attracting (or superattracting) periodic cycle; the hyperbolic maps form an open subset of $\mathcal{M}_d$ whose connected components are the hyperbolic components. Familiar examples are the main cardioid and the bulbs of the Mandelbrot set for quadratic polynomials. By the $J$-stability theory of Mañé–Sad–Sullivan, the maps in one hyperbolic component are topologically conjugate on their Julia sets, and moreover their Julia sets are quasisymmetrically equivalent: a quasisymmetry is a homeomorphism that distorts relative distances in a uniformly controlled way, and on subsets of the sphere these are exactly the boundary restrictions of quasiconformal homeomorphisms. Consequently each hyperbolic component $\mathcal{H}$ carries one common geometric invariant, the conformal dimension $\dim_C(J(f))$ for $f\in\mathcal{H}$, defined as the infimum of Hausdorff dimensions over all metric spaces quasisymmetrically equivalent to $J(f)$, together with the varying function $f\mapsto\dim_H J(f)$, which Ruelle proved is real-analytic on $\mathcal{H}$. It is classical that a hyperbolic rational map has $\dim_H J(f)<2$, so the value $2$ can only appear as a non-attained supremum.

The one general mechanism that forces dimensions close to $2$ is parabolic bifurcation. A periodic point is parabolic if its multiplier is a root of unity; nearby points are attracted to the cycle along finitely many 'petals', and the Fatou components carrying this attraction form the immediate parabolic basin. Shishikura proved in 1998 that if a rational map has a parabolic periodic point whose immediate parabolic basin contains exactly one critical point, then maps obtained from it by small perturbations whose nearby multiplier tends to $1$ along suitable horocyclic directions in parameter space have Julia sets with $\dim_H$ arbitrarily close to $2$. Thus, whenever a hyperbolic component can be degenerated, along a path staying inside it, to such a parabolic map, the supremum of $\dim_H J(f)$ over the component equals $2$.

Qiu and Yang developed this circle of ideas for Cantor circle Julia sets — Julia sets homeomorphic to the product of a Cantor set with a circle, which arise for instance in McMullen's family $z\mapsto z^n+\lambda/z^\ell$. They proved that every hyperbolic Cantor circle Julia set is quasisymmetrically equivalent to a standard model with round-circle components, gave a computable formula for the number $N(d)$ of Cantor circle hyperbolic components in $\mathcal{M}_d$, and showed for every Cantor circle hyperbolic component $\mathcal{H}$ that $\inf_{f\in\mathcal{H}}\dim_H J(f)=\dim_C(J(f_0))=1+\alpha$ (with $\alpha$ determined by the combinatorial data of the component) and $\sup_{f\in\mathcal{H}}\dim_H J(f)=2$. The proof of the supremum statement pinches a map of the component, along a hyperbolic-parabolic deformation in the sense of Cui and Tan, to a parabolic map whose immediate parabolic basin contains exactly one critical point, and then applies Shishikura's theorem. Their Theorem 6.2 isolates what the argument needs and proves $\sup_{f\in\mathcal{H}}\dim_H J(f)=2$ for every hyperbolic component $\mathcal{H}$ (any $d\ge2$) in which every map has a simply connected periodic Fatou component whose closure is disjoint from the other Fatou components; combined with work of Fu and Yang this also covers all Sierpiński carpet components (a Sierpiński carpet being the compact planar set remaining after deleting from a closed disk countably many open subdisks with pairwise disjoint closures, in the pattern of the standard carpet).

Immediately after their Theorem 1.4, Qiu and Yang state: 'we believe that $\sup_{f\in\mathcal{H}}\dim_H(J(f))=2$ holds for any hyperbolic component in the space of rational maps $\operatorname{Rat}_d$ with any $d\ge2$.' The belief is consistent with the broader experience — going back to Haïssinsky and Pilgrim's construction of quasisymmetrically inequivalent hyperbolic Julia sets distinguished by their conformal dimensions — that hyperbolic Julia-set geometry degrades maximally near component boundaries, but no proof is known beyond the class of Theorem 6.2. The gap is concrete: in components whose periodic Fatou components are all multiply connected — for example components of maps with a Herman ring (an annular Fatou component on which an iterate acts as an irrational rotation) or with only infinitely connected attracting basins — no simply connected basin is available from which to run the pinching-plus-bifurcation argument, and no substitute mechanism is known. A companion question of the same paper, whether $\inf_{f\in\mathcal{H}}\dim_H J(f)=\dim_C(J(f_0))$ for every hyperbolic component, is likewise answered only in special classes and remains open in general.

Problem Statement

Let $d\ge2$ and let $\mathcal{H}$ be a hyperbolic component of the moduli space $\mathcal{M}_d$ of rational maps of degree $d$ (equivalently, a component of the hyperbolic locus in $\operatorname{Rat}_d$, taken up to Möbius conjugacy). Prove or refute:

$$\sup_{f\in\mathcal{H}}\ \dim_H J(f)\;=\;2.$$

That is, for every $\varepsilon>0$, does $\mathcal{H}$ contain a rational map $f$ with $\dim_H J(f)>2-\varepsilon$?

The equality is already known for components in which every map has a simply connected periodic Fatou component whose closure is disjoint from the other Fatou components (Qiu–Yang, Theorem 6.2), a class that includes all Cantor circle components and, by the same paper combined with Fu–Yang, all Sierpiński carpet components. The problem therefore concerns the remaining components — in particular those in which no periodic Fatou component is simply connected, such as components of hyperbolic maps with Herman rings or with only multiply connected attracting basins, and more generally any component failing the Theorem 6.2 hypothesis. Since $\dim_H J(f)<2$ for every hyperbolic rational map, the asserted supremum is necessarily a non-attained limit; the question is whether dimension $2$ is approachable within every hyperbolic component. A resolving answer must either cover all degrees $d\ge2$ and all hyperbolic components, or, in the refuting direction, exhibit one specific hyperbolic component together with a rigorous uniform bound $\sup_{f\in\mathcal{H}}\dim_H J(f)\le2-\varepsilon$ for some $\varepsilon>0$.

References

1. Weiyuan Qiu and Fei Yang, Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets, Transactions of the American Mathematical Society 374 (2021) 5191–5223, DOI 10.1090/tran/8392, https://doi.org/10.1090/tran.8392 (preprint arXiv:1811.10042, https://arxiv.org/abs/1811.10042)
2. Yuming Fu and Fei Yang, Area and Hausdorff dimension of Sierpiński carpet Julia sets, Mathematische Zeitschrift 294 (2020) 1441–1456, DOI 10.1007/s00209-019-02319-4, https://doi.org/10.1007/s00209-019-02319-4 (preprint arXiv:1812.03016, https://arxiv.org/abs/1812.03016)
3. M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Annals of Mathematics (2) 147 (1998) 225–267, DOI 10.2307/121009, https://doi.org/10.2307/121009
4. G. Cui and L. Tan, Hyperbolic-parabolic deformations of rational maps, Science China Mathematics 61 (2018) 2157–2220, DOI 10.1007/s11425-018-9426-4, https://doi.org/10.1007/s11425-018-9426-4
5. D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynamical Systems 2 (1982) 99–107, DOI 10.1017/S0143385700009603, https://doi.org/10.1017/S0143385700009603
6. R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Annales scientifiques de l'École normale supérieure (4) 16 (1983) 193–217, DOI 10.24033/asens.1446, https://doi.org/10.24033/asens.1446
7. P. Haïssinsky and K. M. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Revista Matemática Iberoamericana 28 (2012) 1025–1034, DOI 10.4171/rmi/701, https://doi.org/10.4171/rmi/701
8. Xiaole He, Yingqing Xiao and Fei Yang, Rational maps with Cantor bubble Julia sets, preprint (2026), arXiv:2604.20096, https://arxiv.org/abs/2604.20096
9. Yingqing Xiao and Fei Yang, Disconnected Julia sets as buried Julia components, Mathematische Zeitschrift 308 (2024), DOI 10.1007/s00209-024-03631-4, https://doi.org/10.1007/s00209-024-03631-4
10. James Belk and Bradley Forrest, Quasisymmetries of finitely ramified Julia sets, Mathematische Annalen 393 (2025) 1683–1740, DOI 10.1007/s00208-025-03238-y, https://doi.org/10.1007/s00208-025-03238-y
11. Insung Park, Julia sets with Ahlfors-regular conformal dimension one, preprint (2022), arXiv:2209.13384, https://arxiv.org/abs/2209.13384
12. Fei Yang, Cantor Julia sets with Hausdorff dimension two, International Mathematics Research Notices 2021 (2021) 4994–5006, DOI 10.1093/imrn/rny286, https://doi.org/10.1093/imrn/rny286 (preprint arXiv:1802.01063, https://arxiv.org/abs/1802.01063)
13. Gorachand Chakraborty, Subhasis Ghora and Tarakanta Nayak, Herman Rings: Structure, Dynamics, and Open Problems, preprint (2025), arXiv:2512.24118, https://arxiv.org/abs/2512.24118