McMullen conjectured in 1995, as the rational-dynamics counterpart of Thurston's compactness theorem for acylindrical hyperbolic 3-manifolds, that a hyperbolic component of the moduli space of rational maps whose Julia sets are Sierpiński carpets is bounded. The conjecture has been resolved only in special regimes: degree 2 (Epstein), quasi post-critically finite degenerations (Luo), and, as of September 2025, all disjoint-type components in arbitrary degree (Dudko–Luo). The surviving open core is the non-disjoint-type case, where at least two critical points share a periodic cycle of Fatou components, so the interacting critical dynamics currently defeats both the renormalization-based and the extremal-length-based proof strategies.

Background

A rational map f:C^→C^f:\hat{\mathbb{C}}\to\hat{\mathbb{C}} of degree d≥2d\ge 2 splits the Riemann sphere into the Fatou set, where the iterates of ff form a normal family, and its complement, the Julia set J(f)J(f), which carries the chaotic dynamics. The map is called hyperbolic if every critical point converges under iteration to an attracting periodic cycle. Hyperbolic maps form an open, conjecturally dense subset of the moduli space Md,fm\mathcal{M}_{d,\mathrm{fm}} of degree-dd rational maps marked by an ordered list of their d+1d+1 fixed points (the marking removes the ambiguity of conjugation by Möbius transformations); a connected component H\mathcal{H} of this hyperbolic locus is a hyperbolic component. As [f][f] varies within H\mathcal{H} the topological dynamics on J(f)J(f) stays constant while the geometry of J(f)J(f) varies, and H\mathcal{H} need not be bounded in moduli: Makienko constructed unbounded hyperbolic components in general. A compact subset of the sphere is a Sierpiński carpet if it is obtained by removing from the sphere a countable family of open Jordan disks with pairwise disjoint closures whose diameters tend to zero; many hyperbolic rational maps have Sierpiński carpet Julia sets, for example z2+λ/z2z^2+\lambda/z^2 for suitable λ\lambda. Under Sullivan's dictionary between rational iteration and Kleinian groups, convex-cocompact acylindrical Kleinian groups correspond to maps with Sierpiński carpet Julia set (a compact hyperbolic 3-manifold is acylindrical exactly when its limit set is a Sierpinski carpet), and Thurston proved that the deformation space of an acylindrical 3-manifold is bounded. Guided by this analogy, McMullen conjectured that a hyperbolic component H⊆Md,fm\mathcal{H}\subseteq\mathcal{M}_{d,\mathrm{fm}} whose maps have Sierpiński carpet Julia sets is bounded in Md,fm\mathcal{M}_{d,\mathrm{fm}}. A hyperbolic component is of disjoint type if every map in it has 2d−22d-2 distinct attracting periodic cycles, i.e. the critical points lie in pairwise different cycles of periodic Fatou components; in that case the multipliers of the attracting cycles parametrize H≅D2d−2\mathcal{H}\cong\mathbb{D}^{2d-2}. Partial results prior to 2025: Epstein proved boundedness for disjoint-type quadratic hyperbolic components; Nie and Pilgrim proved boundedness for bicritical maps with two distinct attracting cycles of period at least two and for quartic Newton-map components (while showing that some Newton hyperbolic components are unbounded, so the carpet hypothesis is essential); Luo proved that within a Sierpiński carpet hyperbolic component of any type, every quasi post-critically finite degeneration — a sequence leaving every compact subset of H\mathcal{H} whose critical orbits are uniformly quasi-periodic in the hyperbolic metric of the critical and post-critical Fatou components — has a convergent subsequence in moduli. In September 2025, Dudko and Luo resolved the disjoint-type case completely: a Sierpiński carpet hyperbolic component of disjoint type is bounded in every degree, the multiplier parametrization extends to a homeomorphism H‾≅D‾2d−2\overline{\mathcal{H}}\cong\overline{\mathbb{D}}^{2d-2} (so ∂H\partial\mathcal{H} is locally connected), and every non-repelling periodic point of every boundary map admits a quadratic-like restriction of uniformly bounded modulus. Their proof combines the Thurston-type obstruction analysis of limiting dynamics on rescaling trees, which is available for Sierpiński components of any type, with uniform a priori bounds (pseudo-Siegel bounds) from renormalization theory for the geometrically infinite boundary maps; the latter ingredient is exactly what is currently missing when critical points interact within one Fatou component.

Problem Statement

Prove or refute McMullen's boundedness conjecture: every hyperbolic component H⊆Md,fm\mathcal{H}\subseteq\mathcal{M}_{d,\mathrm{fm}} of the moduli space of marked degree-dd rational maps whose maps have Sierpiński carpet Julia sets is bounded in Md,fm\mathcal{M}_{d,\mathrm{fm}}; equivalently, every sequence ([fn])([f_n]) in H\mathcal{H} has a subsequence converging in the moduli space. Since the disjoint-type case (each map of H\mathcal{H} having 2d−22d-2 attracting periodic cycles) is now a theorem of Dudko and Luo, the unresolved instance is the non-disjoint-type case, in which at least two critical points lie in the same periodic cycle of Fatou components: prove boundedness there, or refute the conjecture by exhibiting a non-disjoint-type Sierpiński carpet hyperbolic component together with a sequence in it that diverges in Md,fm\mathcal{M}_{d,\mathrm{fm}}.

References

  1. 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
  2. Dzimitry Dudko and Yusheng Luo, "Sierpinski carpet hyperbolic components of disjoint type are bounded", preprint, 2025. DOI: 10.48550/arXiv.2509.25658; arXiv:2509.25658, https://arxiv.org/abs/2509.25658
  3. C. T. 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, https://doi.org/10.4310/cdm.1995.v1995.n1.a4
  4. Adam Lawrence Epstein, "Bounded hyperbolic components of quadratic rational maps", Ergodic Theory and Dynamical Systems 20 (2000) 727–748. DOI: 10.1017/S0143385700000390, https://doi.org/10.1017/S0143385700000390
  5. 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
  6. Hongming Nie and Kevin M. Pilgrim, "Bounded hyperbolic components of bicritical rational maps", Journal of Modern Dynamics 18 (2022) 533–553. DOI: 10.3934/jmd.2022016; arXiv:1903.08873, https://arxiv.org/abs/1903.08873
  7. Hongming Nie and 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
  8. J. Kahn, "Hyperbolic Components and Limits of Extremal Length", MSRI talks, Spring 2022, https://vimeo.com/704224013, https://vimeo.com/700148806