McMullen's 1995 conjecture (Question 5.3 of 'The classification of conformal dynamical systems') predicts that a hyperbolic component of the space of rational maps of degree dd whose Julia sets are Sierpiński carpets is bounded — equivalently relatively compact — in the moduli space of rational maps, in analogy with Thurston's compactness theorem for acylindrical hyperbolic 3-manifolds under Sullivan's dictionary. The disjoint-type case (each critical point attracted along its own periodic Fatou cycle) is now settled: Epstein proved the quadratic case, and Dudko–Luo (2025) proved all degrees, the first boundedness theorem for an entire hyperbolic component in degree d≥3d\ge 3. Luo (2022) had previously settled boundedness along all quasi post-critically finite degenerations. The conjecture remains open for components of non-disjoint (capture) type, in which a periodic cycle of Fatou components contains several critical points; this is the surviving open core.

Background

A rational map is a holomorphic self-map f:C^→C^f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}} of the Riemann sphere given by a ratio of two polynomials, of degree d≥2d\ge 2. Under iteration the sphere splits into the Fatou set, the largest open set on which the iterates form a normal family (every sequence of iterates has a subsequence converging uniformly on compact subsets — the locus of stable behavior), and its complement, the Julia set J(f)J(f), the locus of chaotic behavior. The map is called hyperbolic (in McMullen's earlier terminology, expanding) if every critical point of ff converges under iteration to an attracting periodic cycle. Hyperbolic maps form an open subset of the moduli space Md=Ratd/PSL2(C)M_d=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C}) of degree-dd rational maps modulo Möbius conjugacy, and a connected component of this open set is a hyperbolic component. Topological dynamics is constant within a hyperbolic component, so any dynamical-topological property of the Julia set, such as being a Sierpiński carpet, holds either for all maps in the component or for none.

A Sierpiński carpet is any set homeomorphic to the standard carpet obtained from the sphere by removing a countable dense family of open Jordan disks with pairwise disjoint closures (Whyburn's topological characterization). For a hyperbolic rational map the Julia set is a Sierpiński carpet exactly when it is connected and all Fatou components are simply connected domains whose closures are pairwise disjoint. Hyperbolic components are classified by how critical points sit in the Fatou set: a component is of disjoint type if all critical points lie in pairwise different periodic cycles of Fatou components (equivalently, its maps have 2d−22d-2 distinct attracting cycles); otherwise a periodic cycle of Fatou components captures several critical points, a configuration often called a capture or shared-cycle component. A hyperbolic component has a distinguished center, the map whose critical points are all preperiodic (eventually periodic); McMullen's question is phrased in terms of this center.

The question is motivated by the dictionary between rational maps and Kleinian groups (Sullivan's dictionary), the systematic analogy between holomorphic iteration and hyperbolic geometry; a Kleinian group is a discrete group of Möbius transformations acting on the sphere and on hyperbolic 3-space, and its limit set is the accumulation set of any orbit. A compact hyperbolic 3-manifold whose boundary surfaces are incompressible (their fundamental groups inject into that of the manifold) is called acylindrical when every essential cylinder in it can be homotoped into the boundary, and this holds exactly when the limit set of its Kleinian group is a Sierpiński curve; Thurston proved that deformation spaces of acylindrical 3-manifolds are compact. Since acylindrical Kleinian groups have carpet limit sets, McMullen asked in 1995 for the rational-map analogue: are hyperbolic components with carpet Julia sets compact in moduli space? The carpet hypothesis is essential — unbounded hyperbolic components are known to exist in general (Epstein's quadratic criterion implies that components one of whose attractors is a fixed point fail to have compact closure, and Makienko and Tan Lei constructed further unbounded components), and the question is which dynamical conditions force boundedness. Beyond its intrinsic parameter-space interest, boundedness of carpet components is a compactness ingredient for the realization and rigidity program for rational maps — Thurston's characterization of rational maps among branched coverings, and the combination operations known as matings and tunings — and a proof must exclude every degenerating sequence, including those whose boundary limits are geometrically infinite maps (maps whose postcritical dynamics admits no finite combinatorial description, the rational-map analogue of geometrically infinite Kleinian groups).

Problem Statement

State and prove, or refute, McMullen's conjecture (Question 5.3 of 'The classification of conformal dynamical systems', 1995) in its natural generality: let d≥2d\ge 2 and let H\mathcal{H} be a hyperbolic component in the moduli space Md=Ratd/PSL2(C)M_d=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C}) of rational maps of degree dd (equivalently, in any fixed standard finite marking of the maps, such as the fixed-point-marked moduli space Md,fmM_{d,\mathrm{fm}}; boundedness is equivalent under the change of marking). Suppose the Julia set of the maps in H\mathcal{H} is a Sierpiński carpet. Is H\mathcal{H} bounded — equivalently, relatively compact — in moduli space? Concretely: prove that no sequence of maps in H\mathcal{H} can escape to the boundary of moduli space (for instance by degenerating towards a map of lower degree), with no restriction on the type of the component, i.e. allowing periodic Fatou cycles that contain several critical points (non-disjoint or capture configurations) alongside the disjoint-type case. The disjoint-type case is already a theorem (Epstein in degree 2; Dudko–Luo in all degrees), so the open content of the conjecture is precisely the components of non-disjoint type.

References

  1. C. McMullen, The classification of conformal dynamical systems, Current Developments in Mathematics 1995, International Press, pp. 323–360. DOI 10.4310/cdm.1995.v1995.n1.a4; author PDF at https://people.math.harvard.edu/~ctm/home/text/papers/curdev/curdev.pdf
  2. Adam Epstein, Bounded hyperbolic components of quadratic rational maps, Ergodic Theory and Dynamical Systems 20 (2000), no. 3, 727–748. DOI 10.1017/S0143385700000390, https://doi.org/10.1017/S0143385700000390
  3. Yusheng Luo, On geometrically finite degenerations I: boundaries of main hyperbolic components, Journal of the European Mathematical Society 26 (2024), no. 10, 3793–3861. DOI 10.4171/jems/1342; arXiv:2101.07880, https://arxiv.org/abs/2101.07880
  4. Yusheng Luo, On geometrically finite degenerations II: convergence and divergence, Transactions of the American Mathematical Society 375 (2022), 3469–3527. DOI 10.1090/tran/8597; arXiv:2102.00357, https://arxiv.org/abs/2102.00357
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  6. Hongming Nie, Kevin M. Pilgrim, Boundedness of hyperbolic components of Newton maps, Israel Journal of Mathematics 238 (2020), 837–869. DOI 10.1007/s11856-020-2044-6; arXiv:1809.02722, https://arxiv.org/abs/1809.02722
  7. Peter Makienko, Unbounded components in parameter space of rational maps, Conformal Geometry and Dynamics 4 (2000), 1–21. DOI 10.1090/S1088-4173-00-00044-8, https://doi.org/10.1090/S1088-4173-00-00044-8
  8. Lei Tan, On pinching deformations of rational maps, Annales Scientifiques de l'École Normale Supérieure (4) 35 (2002), no. 3, 353–370. DOI 10.1016/S0012-9593(02)01092-3, https://doi.org/10.1016/S0012-9593(02)01092-3
  9. Dzimitry Dudko, Yusheng Luo, Sierpiński carpet hyperbolic components of disjoint type are bounded, preprint, 2025. arXiv:2509.25658, https://arxiv.org/abs/2509.25658
  10. Dzmitry Dudko, On the MLC Conjecture and the Renormalization Theory in Complex Dynamics, survey, 2025, to appear in the Proceedings of the ICM 2026. arXiv:2512.24171, https://arxiv.org/abs/2512.24171