McMullen asked (Current Developments in Mathematics, 1995, Question 5.3), by analogy with Thurston's compactness theorem for acylindrical hyperbolic 3-manifolds, whether a hyperbolic component of the moduli space of rational maps of degree dd whose maps have Sierpiński carpet Julia sets is compact. The conjecture is confirmed for degree-two components whose maps have two distinct attracting cycles, neither a fixed point (Epstein), for strict type-D bicritical components of all degrees (Nie–Pilgrim), and, very recently, for all components of disjoint type in every degree (Dudko–Luo, 2025); it remains open for carpet hyperbolic components not of disjoint type: capture-type components (some critical point in a strictly preperiodic Fatou domain, already arising in degree two) and components in which several critical points converge to a shared attracting cycle — the latter certainly arise in degrees d≥3d\geq 3, and in degree two they are the bitransitive components, for which carpet Julia sets are excluded only when the attracting period is at most four (Devaney–Fagella–Garijo–Jarque), bitransitive carpet examples having been constructed by Look in the family z↦λ(z+1/z)z\mapsto\lambda(z+1/z).

Background

A rational map is a holomorphic self-map ff of the Riemann sphere C^\widehat{\mathbb{C}} of degree d≥2d\geq 2; the sphere splits into the Fatou set (the open set where the iterates fnf^n form a normal family, i.e. every sequence of iterates has a locally uniformly convergent subsequence) and its complement, the Julia set J(f)J(f), on which the dynamics is chaotic. The map is called hyperbolic (McMullen says expanding) if every critical point of ff converges under iteration to an attracting periodic cycle. The space Ratd\mathrm{Rat}_d of all rational maps of degree dd is a complex affine variety, and the hyperbolic maps form an open subset whose connected components are the hyperbolic components; since hyperbolicity is preserved by holomorphic conjugation, one studies components in the moduli space Vd=Ratd/PSL2(C)V_d=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C}) of conjugacy classes (conjugation itself degenerates inside any component, so boundedness is a question about VdV_d, or equivalently about the fixed-point-marked moduli space Md,fm=Ratd,fm/PSL2(C)M_{d,\mathrm{fm}}=\mathrm{Rat}_{d,\mathrm{fm}}/\mathrm{PSL}_2(\mathbb{C}), obtained by labeling the d+1d+1 fixed points of each map, a finite cover of VdV_d). Within a hyperbolic component the topological dynamics on the Julia set is constant, and when J(f)J(f) is connected the component contains a unique postcritically finite map f0f_0 (all critical orbits finite), called its center. The Julia set of a hyperbolic rational map is a Sierpiński carpet (McMullen: Sierpiński curve) — the sphere with a countable dense family of open disks with pairwise disjoint Jordan-curve boundaries removed — exactly when the boundaries of all Fatou domains are pairwise disjoint Jordan curves; many examples are known, e.g. the quadratic rational maps classified by Devaney–Fagella–Garijo–Jarque and the bitransitive examples of Look in the family z↦λ(z+1/z)z\mapsto\lambda(z+1/z). Not all hyperbolic components can be bounded: results of Makienko and Tan Lei (as surveyed by Nie–Pilgrim and by Dudko–Luo) show that components with an attracting fixed point, and certain components with both critical points in one immediate basin, are unbounded in moduli space. The carpet case is the acylindrical case of Sullivan's dictionary, the systematic analogy between iterated rational maps and Kleinian groups (discrete groups of Möbius transformations acting on the sphere): a convex-cocompact Kleinian group is acylindrical — its quotient hyperbolic 3-manifold contains no essential annuli — precisely when its limit set is a Sierpiński carpet, and Thurston proved that deformation spaces of acylindrical hyperbolic 3-manifolds are compact. McMullen's Question 5.3 predicts the exact rational-map counterpart: the Sierpiński carpet condition on the Julia set should force compactness of the component. When McMullen posed it, not a single expanding component was known to have compact closure; Dudko–Luo (2025) describe the general conjecture as one of the bottlenecks for the realization and rigidity theory of rational maps.

Problem Statement

McMullen's Question 5.3, in the source-faithful modern formulation (Dudko–Luo's Conjecture 1.1): let HH be a hyperbolic component of the moduli space Vd=Ratd/PSL2(C)V_d=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C}) of rational maps of degree d≥2d\geq 2 (equivalently, of the fixed-point-marked moduli space Md,fmM_{d,\mathrm{fm}}), and suppose the Julia set of one — equivalently every — map in HH is a Sierpiński carpet. Is HH bounded, i.e. relatively compact, in VdV_d? Equivalently, in McMullen's original wording: if the expanding component UU has center f0f_0 and J(f0)J(f_0) is a Sierpiński curve, is U⊂VdU\subset V_d compact? Boundedness is proved for degree-two components whose maps have two distinct attracting cycles, neither a fixed point (Epstein); for bicritical strict type D components of any degree (Nie–Pilgrim); and for disjoint-type carpet components of any degree (Dudko–Luo). The question remains open for the Sierpiński carpet hyperbolic components that are not of disjoint type, namely capture-type components (some critical point in a strictly preperiodic Fatou domain, including the degree-two components certified by Devaney–Fagella–Garijo–Jarque) and components in which several critical points converge to the same attracting cycle — such shared-cycle components necessarily arise in degrees d≥3d\geq 3, and in degree two they are the bitransitive (type B) components, for which carpet Julia sets are excluded only when the attracting period is at most four (Devaney–Fagella–Garijo–Jarque, Theorem B(b)) and for which bitransitive carpet examples are constructed by Look in the family z↦λ(z+1/z)z\mapsto\lambda(z+1/z). Either a proof of boundedness for all such components or an explicit carpet hyperbolic component shown to be unbounded in moduli space resolves the question.

References

  1. C. T. McMullen, The classification of conformal dynamical systems, Current Developments in Mathematics 1995, International Press, pp. 323–360 (Question 5.3), https://people.math.harvard.edu/~ctm/home/text/papers/curdev/curdev.pdf
  2. A. 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. R. L. Devaney, N. Fagella, A. Garijo, X. Jarque, Sierpinski curve Julia sets for quadratic rational maps, Annales Academiae Scientiarum Fennicae Mathematica 39 (2014), 3–22, DOI 10.5186/aasfm.2014.3903, https://doi.org/10.5186/aasfm.2014.3903
  4. H. Nie, K. M. Pilgrim, Bounded hyperbolic components of bicritical rational maps, Journal of Modern Dynamics 18 (2022), 533–553, DOI 10.3934/jmd.2022016 (preprint arXiv:1903.08873), https://arxiv.org/abs/1903.08873
  5. Y. Gao, L. Yang, J. Zeng, On rational maps with buried critical points, arXiv:2002.03544 (2020); published in abridged form as Subhyperbolic rational maps on boundaries of hyperbolic components, Discrete Contin. Dyn. Syst. 42 (2022), 319–326, DOI 10.3934/dcds.2021118, https://arxiv.org/abs/2002.03544
  6. D. Dudko, Y. Luo, Sierpinski carpet hyperbolic components of disjoint type are bounded, arXiv:2509.25658 (2025), https://arxiv.org/abs/2509.25658
  7. D. M. Look, An infinite sequence of bitransitive Sierpiński carpets for z↦λ(z+1/z)z\mapsto\lambda(z+1/z), International Journal of Bifurcation and Chaos 32 (2022), no. 3, Article 2250037, DOI 10.1142/S0218127422500377, https://doi.org/10.1142/S0218127422500377