In the moduli space of marked degree-dd rational maps, every hyperbolic component of disjoint type is biholomorphic to the polydisc D2d−2\mathbb{D}^{2d-2} via the multipliers of its 2d−22d-2 attracting cycles, but such components need not be bounded: parameter sequences can escape to infinity while their multipliers converge to points of ∂D2d−2\partial\mathbb{D}^{2d-2}. Dudko and Luo, having proved that Sierpiński carpet components of disjoint type are bounded, formulated Conjecture 1.9: for a general hyperbolic component H\mathcal{H} of disjoint type, the obstructed boundary at infinity ∂∞H\partial^{\infty}\mathcal{H} — the set of multiplier limits of sequences in H\mathcal{H} escaping to infinity in moduli — equals the closure of its rational part ∂Q∞H\partial^{\infty}_{\mathbb{Q}}\mathcal{H}, where only rational indifferent multipliers e2πip/qe^{2\pi i p/q} occur. This would make the degeneration at infinity of every disjoint-type component effectively computable from combinatorial (rational) obstructions. The conjecture is verified in the source only for the prototype component Hz2\mathcal{H}_{z^2} of z↦z2z\mapsto z^2; the authors' announced sequel is to supply the key combinatorial bound on the pulled-off constant NSiegelN_{Siegel}, and the conjecture remains open in general.

Background

A rational map f:P1→P1f:\mathbb{P}^1\to\mathbb{P}^1 of degree d≥2d\geq 2 splits the sphere into the Fatou set (where iterates form a normal family) and its complement, the Julia set. The map is called hyperbolic if all critical points converge to attracting cycles; the number of independent attracting cycles is at most 2d−22d-2. One works in the moduli space Md,fm\mathcal{M}_{d,\mathrm{fm}} of fixed-point marked rational maps — rational maps of degree dd with all d+1d+1 fixed points marked, modulo Möbius conjugacy — in which the multiplier of each marked fixed point is a well-defined holomorphic function. Hyperbolic maps form an open set, and a connected component H\mathcal{H} of this set is a hyperbolic component. Two refinements matter here: H\mathcal{H} is of disjoint type if each of its maps has exactly 2d−22d-2 attracting cycles whose immediate basins (the Fatou components containing those cycles) are pairwise disjoint, and it is a Sierpiński carpet component if the Julia set of its maps is a Sierpiński carpet (a plane carpet obtained by removing countably many open squares with disjoint closures; the generic Fatou geometry for disjoint-type maps of sufficiently high type). For a disjoint-type component, the multipliers of the 2d−22d-2 attracting cycles give a biholomorphism ρ:H→≃D2d−2\boldsymbol{\rho}:\mathcal{H}\xrightarrow{\simeq}\mathbb{D}^{2d-2} onto a polydisc, so the component carries an intrinsic 'unit polydisc' compactification; the geometric question is how H\mathcal{H} sits in Md,fm\mathcal{M}_{d,\mathrm{fm}}, and in particular whether it is bounded there. Motivated by Thurston's compactness theorem for acylindrical hyperbolic 3-manifolds (compact 3-manifolds whose essential annuli are all boundary-parallel) under Sullivan's dictionary between Kleinian groups and rational maps, McMullen conjectured in the early 1990s (Conjecture 1.1 of the source paper) that Sierpiński carpet hyperbolic components are bounded; this stayed open beyond degree 2 for decades and was a stated bottleneck for the realization and rigidity theory of rational maps. Dudko and Luo (2025) proved it in the disjoint-type case: a Sierpiński carpet component of disjoint type is bounded, the multiplier map extends to a homeomorphism H‾≅D‾2d−2\overline{\mathcal{H}}\cong\overline{\mathbb{D}}^{2d-2}, and every map on H‾\overline{\mathcal{H}} admits a quadratic-like restriction (a holomorphic conjugacy between nested topological disks, as in Douady–Hubbard polynomial-like theory) with uniformly bounded modulus around each non-repelling periodic point. The proof rests on uniform a priori geometric bounds (Theorem C of the source) that control postcritical sets (the forward orbits of the critical points) of boundary maps in terms of a combinatorial 'pulled-off' constant NSiegel([f])N_{Siegel}([f]), which measures Siegel-type degeneration: a Siegel disk is a Fatou component on which the map is holomorphically conjugate to an irrational rotation, and NSiegelN_{Siegel} quantifies how such disks degenerate along a family. This constant is uniformly bounded for the eventually-golden-mean boundary combinatorics — boundary maps whose indifferent multipliers are e2πiθe^{2\pi i\theta} with θ\theta a rotation number whose continued-fraction expansion is eventually all 11s — the class available in the Sierpiński case. For general disjoint-type components, which may be unbounded in moduli even though ρ\boldsymbol{\rho} identifies them with D2d−2\mathbb{D}^{2d-2}, Dudko and Luo parameterize the escape to infinity by Definition 1.8: the obstructed boundary at infinity is ∂∞H={ρ∈∂D2d−2:∃[fn]∈H\partial^{\infty}\mathcal{H}=\{\rho\in\partial\mathbb{D}^{2d-2}:\exists [f_n]\in\mathcal{H} with [fn]→∞[f_n]\to\infty in Md,fm\mathcal{M}_{d,\mathrm{fm}} and ρ([fn])→ρ}\boldsymbol{\rho}([f_n])\to\rho\}, and its rational part ∂Q∞H=∂∞H∩∂QD2d−2\partial^{\infty}_{\mathbb{Q}}\mathcal{H}=\partial^{\infty}\mathcal{H}\cap\partial_{\mathbb{Q}}\mathbb{D}^{2d-2} consists of those limits whose indifferent multipliers are all roots of unity e2πip/qe^{2\pi i p/q}; the rational part is identifiable with obstructed geometrically finite maps and is effectively computable. The boundary ∂H\partial\mathcal{H} splits into a regular part ∂regH\partial_{\mathrm{reg}}\mathcal{H} (maps with exactly 2d−22d-2 non-repelling periodic cycles, on which the extended multiplier map embeds into ∂D2d−2\partial\mathbb{D}^{2d-2} by transversality) and an exceptional part ∂excH\partial_{\mathrm{exc}}\mathcal{H} (where at least two non-repelling cycles collide; rigidity can fail there, e.g. the exceptional boundary of Hz2\mathcal{H}_{z^2} fibers over the multiplier pair (1,1)(1,1)). In the prototype case of the component Hz2⊂M2,fm\mathcal{H}_{z^2}\subset\mathcal{M}_{2,\mathrm{fm}} containing z↦z2z\mapsto z^2 — disjoint type but not Sierpiński, and unbounded — the source computes ∂∞Hz2={(e2πit,e−2πit)}=∂Q∞Hz2‾\partial^{\infty}\mathcal{H}_{z^2}=\{(e^{2\pi i t},e^{-2\pi i t})\}=\overline{\partial^{\infty}_{\mathbb{Q}}\mathcal{H}_{z^2}}, verifying the desired characterization in that example.

Problem Statement

Prove or refute Conjecture 1.9 of Dudko–Luo (arXiv:2509.25658). Let H⊆Md,fm\mathcal{H}\subseteq\mathcal{M}_{d,\mathrm{fm}} be a hyperbolic component of disjoint type in the moduli space of fixed-point marked degree-dd rational maps, d≥2d\geq 2, with multiplier map ρ:H→≃D2d−2\boldsymbol{\rho}:\mathcal{H}\xrightarrow{\simeq}\mathbb{D}^{2d-2}. Define the obstructed boundary at infinity ∂∞H={ρ∈∂D2d−2: ∃ [fn]∈H with [fn]→∞ in Md,fm and ρ([fn])→ρ}\partial^{\infty}\mathcal{H}=\{\rho\in\partial\mathbb{D}^{2d-2}:\ \exists\,[f_n]\in\mathcal{H}\ \text{with}\ [f_n]\to\infty\ \text{in}\ \mathcal{M}_{d,\mathrm{fm}}\ \text{and}\ \boldsymbol{\rho}([f_n])\to\rho\} and the rational obstructed boundary ∂Q∞H=∂∞H∩∂QD2d−2\partial^{\infty}_{\mathbb{Q}}\mathcal{H}=\partial^{\infty}\mathcal{H}\cap\partial_{\mathbb{Q}}\mathbb{D}^{2d-2}, where ∂QD2d−2\partial_{\mathbb{Q}}\mathbb{D}^{2d-2} consists of boundary tuples whose indifferent multipliers are all rational, i.e. of the form e2πip/qe^{2\pi i p/q}. The conjecture asserts ∂∞H=∂Q∞H‾.\partial^{\infty}\mathcal{H}=\overline{\partial^{\infty}_{\mathbb{Q}}\mathcal{H}}. In particular, the natural extension of the multiplier map is required to identify the regular boundary as ∂regH≅∂D2d−2−∂Q∞H‾−ρ(∂excH)\partial_{\mathrm{reg}}\mathcal{H}\cong\partial\mathbb{D}^{2d-2}-\overline{\partial^{\infty}_{\mathbb{Q}}\mathcal{H}}-\boldsymbol{\rho}(\partial_{\mathrm{exc}}\mathcal{H}), where ∂regH\partial_{\mathrm{reg}}\mathcal{H} is the set of boundary maps with exactly 2d−22d-2 non-repelling periodic cycles and ∂excH\partial_{\mathrm{exc}}\mathcal{H} its complement in ∂H\partial\mathcal{H}. The statement must hold for every disjoint-type component and every degree, with no restriction to Sierpiński components, specific degrees, or restricted rotation combinatorics.

References

  1. Dzimitry Dudko and Yusheng Luo. Sierpinski carpet hyperbolic components of disjoint type are bounded. arXiv:2509.25658, 2025. DOI: 10.48550/arXiv.2509.25658. https://arxiv.org/abs/2509.25658 (source of the problem statement)
  2. Dzmitry Dudko, Yusheng Luo, and Mikhail Lyubich. Uniform a priori bounds for neutral renormalization. Variation II: ψ∙\psi^{\bullet}-ql Siegel maps. arXiv:2509.23031, 2025. https://arxiv.org/abs/2509.23031
  3. Dzmitry Dudko. On the MLC Conjecture and the Renormalization Theory in Complex Dynamics. Proceedings of the International Congress of Mathematicians 2026, Volume 5 (Invited Lectures, Sections 9–11), pages 116–135. SIAM, 2026. DOI: 10.1137/25m1807101. arXiv:2512.24171. https://arxiv.org/abs/2512.24171
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  5. Sabyasachi Mukherjee and S. Viswanathan. Correspondences on hyperelliptic surfaces, combination theorems, and Hurwitz spaces. arXiv:2508.18711, 2025. https://arxiv.org/abs/2508.18711