For a hyperbolic component H\mathcal{H} of disjoint type in the fixed-point-marked moduli space Md,fm\mathcal{M}_{d,\mathrm{fm}} of rational maps, Dudko and Luo conjecture (Conjecture 1.9 of arXiv:2509.25658) that the obstructed boundary at infinity of the multiplier polydisc equals the closure of its rational part: ∂∞H=∂∞QH‾\partial_\infty\mathcal{H}=\overline{\partial^{\mathbb{Q}}_\infty\mathcal{H}}, where ∂∞QH\partial^{\mathbb{Q}}_\infty\mathcal{H} consists of the multiplier limits of sequences escaping to infinity in moduli space whose indifferent coordinates are all rational rotations e2πip/qe^{2\pi ip/q}. The closure is essential: for the quadratic prototype Hz2\mathcal{H}_{z^2}, the paper computes ∂∞Hz2={(e2πit,e−2πit)}\partial_\infty\mathcal{H}_{z^2}=\{(e^{2\pi it},e^{-2\pi it})\}, which equals the closure of the rational points {(e2πip/q,e−2πip/q)}\{(e^{2\pi ip/q},e^{-2\pi ip/q})\} but strictly contains them, so irrational-angle points genuinely occur, though only as limits of rational ones. The authors identify the missing ingredient as a combinatorial bound on the pulled-off constant NSiegel([f])N_{\mathrm{Siegel}}([f]) in terms of the combinatorial distance from ρ([f])\rho([f]) to ∂∞QH‾\overline{\partial^{\mathbb{Q}}_\infty\mathcal{H}}, promised in the sequel announced by the authors. The Sierpinski carpet case is settled vacuously (Theorem A: such components are bounded, so ∂∞H=∅\partial_\infty\mathcal{H}=\emptyset); the general conjecture remains open.

Background

A rational map is a holomorphic self-map f:C^→C^f:\hat{\mathbb{C}}\to\hat{\mathbb{C}} of the Riemann sphere of degree d≥2d\ge2; it has 2d−22d-2 critical points counted with multiplicity. The iterates of ff split the sphere into the Fatou set, where the family of iterates is equicontinuous (regular behavior), and its complement, the Julia set, where the chaotic dynamics concentrates. The map ff is hyperbolic if every critical orbit converges to an attracting cycle, and of disjoint type if, in addition, all critical points lie in pairwise distinct periodic cycles of Fatou components — equivalently, ff has exactly 2d−22d-2 attracting periodic cycles. Hyperbolic maps form open subsets of parameter space whose connected components H\mathcal{H} are the hyperbolic components. After quotienting by Möbius conjugacy and marking the d+1d+1 fixed points, one obtains the fixed-point-marked moduli space Md,fm\mathcal{M}_{d,\mathrm{fm}} of complex dimension 2d−22d-2. On a hyperbolic component of disjoint type, the multiplier map ρ\rho, recording the 2d−22d-2 attracting multipliers (the derivatives of the first-return maps along the attracting cycles), is a biholomorphism ρ:H→≃D2d−2\rho:\mathcal{H}\xrightarrow{\simeq}\mathbb{D}^{2d-2} onto the polydisc, since all attracting multipliers have modulus strictly less than 1. Dudko and Luo study how such components degenerate. Their Definition 1.8 sets ∂∞H={ρ∈∂D2d−2:∃[fn]∈H, [fn]→∞ in Md,fm, ρ([fn])→ρ}\partial_\infty\mathcal{H}=\{\rho\in\partial\mathbb{D}^{2d-2}:\exists[f_n]\in\mathcal{H},\ [f_n]\to\infty\text{ in }\mathcal{M}_{d,\mathrm{fm}},\ \rho([f_n])\to\rho\}, the boundary at infinity: multiplier limits of sequences that stay in H\mathcal{H} but escape to infinity in moduli space (degree fixed, coefficients unbounded). Writing ∂QD2d−2\partial_{\mathbb{Q}}\mathbb{D}^{2d-2} for the boundary tuples all of whose indifferent (unit-modulus) multipliers are rational rotations e2πip/qe^{2\pi ip/q}, the rational obstructed boundary is ∂∞QH=∂∞H∩∂QD2d−2\partial^{\mathbb{Q}}_\infty\mathcal{H}=\partial_\infty\mathcal{H}\cap\partial_{\mathbb{Q}}\mathbb{D}^{2d-2}; the authors note it can be identified with obstructed geometrically finite maps on the boundary of H\mathcal{H} and effectively computed. They further split ∂H=∂regH⊔∂excH\partial\mathcal{H}=\partial^{\mathrm{reg}}\mathcal{H}\sqcup\partial^{\mathrm{exc}}\mathcal{H} (Definition 1.7) into a regular part, where exactly 2d−22d-2 non-repelling periodic cycles persist without collisions, and an exceptional part, where at least two non-repelling cycles collide. The question is the natural higher-dimensional analogue of the classical structure of the Mandelbrot set (the connectedness locus of quadratic polynomials z2+cz^2+c), whose main hyperbolic component has a circle boundary parametrized by internal arguments, with rational internal arguments corresponding to parabolic parameters. McMullen conjectured in the early 1990s that Sierpinski carpet hyperbolic components are bounded (a Sierpinski carpet is a planar set homeomorphic to the standard carpet; disjoint-type maps whose Fatou components are all simply connected with disjoint closures have carpet Julia sets); Dudko and Luo's Theorem A proves this in the disjoint-type case, extending the multiplier map to H‾≅D2d−2‾\overline{\mathcal{H}}\cong\overline{\mathbb{D}^{2d-2}}, so that ∂∞H=∅\partial_\infty\mathcal{H}=\emptyset there and Conjecture 1.9 holds vacuously for Sierpinski components. For the quadratic prototype Hz2\mathcal{H}_{z^2} containing z↦z2z\mapsto z^2, parametrized by the three fixed-point multipliers (ρ1,ρ2,ρ3)(\rho_1,\rho_2,\rho_3) under the holomorphic index constraint ρ1ρ2ρ3−(ρ1+ρ2+ρ3)+2=0\rho_1\rho_2\rho_3-(\rho_1+\rho_2+\rho_3)+2=0, a simple computation in Section 1.5 of the source gives ∂∞Hz2={(e2πit,e−2πit)}={(e2πip/q,e−2πip/q)}‾\partial_\infty\mathcal{H}_{z^2}=\{(e^{2\pi it},e^{-2\pi it})\}=\overline{\{(e^{2\pi ip/q},e^{-2\pi ip/q})\}} — irrational-angle points occur, but only as limits of rational ones; Theorem D of the source is the companion result that the postcritical sets of bi-neutral quadratics (boundary maps with two neutral fixed points) are separated by an annulus whose modulus depends only on the combinatorial distance between the two rotation numbers. For the general case, the authors isolate one missing ingredient. An eventually-golden-mean map is a boundary map [f]∈∂H[f]\in\partial\mathcal{H} whose indifferent periodic cycles have multipliers e2πiθe^{2\pi i\theta} with θ\theta having continued fraction expansion [0;a1,a2,… ][0;a_1,a_2,\dots] with am=1a_m=1 for all large mm (eventually the golden mean). For such maps, the pulled-off constant NSiegel([f])N_{\mathrm{Siegel}}([f]) (Definition 1.5) is the supremum over non-peripheral arcs connecting boundaries of Siegel disks (Fatou components on which an iterate acts as an irrational rotation) of the minimal length of a pull-back sequence of arcs; it measures the combinatorial complexity of Siegel captures along critical orbits. The authors' Theorem C (a priori bounds) controls the degeneration of eventually-golden-mean maps in terms of NSiegel([f])N_{\mathrm{Siegel}}([f]) and the attracting multipliers, and their Pulled-off Principle (Theorem 4.1) bounds NSiegelN_{\mathrm{Siegel}} uniformly only for Sierpinski carpet components — for general disjoint-type components they explicitly warn that NSiegelN_{\mathrm{Siegel}} may be unbounded. They state that Conjecture 1.9 should follow from a bound on NSiegel([f])N_{\mathrm{Siegel}}([f]) depending only on the combinatorial distance (the difference of rotation numbers on the unit circle) between ρ([f])\rho([f]) and ∂∞QH‾\overline{\partial^{\mathbb{Q}}_\infty\mathcal{H}}, and announce a sequel, 'Realization of the boundaries of disjoint type hyperbolic components' (manuscript in preparation), to supply it.

Problem Statement

Let H⊂Md,fm\mathcal{H}\subset\mathcal{M}_{d,\mathrm{fm}} be a hyperbolic component of disjoint type in the fixed-point-marked moduli space of degree-dd rational maps (all critical points in pairwise distinct periodic cycles of Fatou components; equivalently, exactly 2d−22d-2 attracting periodic cycles). Let ρ:H→≃D2d−2\rho:\mathcal{H}\xrightarrow{\simeq}\mathbb{D}^{2d-2} be the multiplier map recording the 2d−22d-2 attracting multipliers, let ∂∞H={ρ∈∂D2d−2:∃[fn]∈H, [fn]→∞ in Md,fm, ρ([fn])→ρ}\partial_\infty\mathcal{H}=\{\rho\in\partial\mathbb{D}^{2d-2}:\exists[f_n]\in\mathcal{H},\ [f_n]\to\infty\text{ in }\mathcal{M}_{d,\mathrm{fm}},\ \rho([f_n])\to\rho\} be the obstructed boundary at infinity, let ∂QD2d−2\partial_{\mathbb{Q}}\mathbb{D}^{2d-2} be the set of boundary tuples all of whose indifferent (unit-modulus) multipliers are rational rotations e2πip/qe^{2\pi ip/q}, and let ∂∞QH=∂∞H∩∂QD2d−2\partial^{\mathbb{Q}}_\infty\mathcal{H}=\partial_\infty\mathcal{H}\cap\partial_{\mathbb{Q}}\mathbb{D}^{2d-2} be the rational obstructed boundary. Conjecture (Dudko–Luo, arXiv:2509.25658, Conjecture 1.9): ∂∞H=∂∞QH‾\partial_\infty\mathcal{H}=\overline{\partial^{\mathbb{Q}}_\infty\mathcal{H}}, the closure taken in ∂D2d−2\partial\mathbb{D}^{2d-2}; in particular, the natural extension of the multiplier map identifies the regular boundary as ∂regH≅∂D2d−2∖∂∞QH‾∖ρ(∂excH)\partial^{\mathrm{reg}}\mathcal{H}\cong\partial\mathbb{D}^{2d-2}\setminus\overline{\partial^{\mathbb{Q}}_\infty\mathcal{H}}\setminus\rho(\partial^{\mathrm{exc}}\mathcal{H}), with ∂regH\partial^{\mathrm{reg}}\mathcal{H} and ∂excH\partial^{\mathrm{exc}}\mathcal{H} the regular and exceptional parts of ∂H\partial\mathcal{H} from Definition 1.7 of the source. Prove or refute this for an arbitrary hyperbolic component of disjoint type with ∂∞H≠∅\partial_\infty\mathcal{H}\neq\emptyset; a component with ∂∞H=∅\partial_\infty\mathcal{H}=\emptyset satisfies the identity vacuously and does not count as a resolution. The Sierpinski carpet case (settled by Theorem A of the source, since boundedness gives ∂∞H=∅\partial_\infty\mathcal{H}=\emptyset) and the quadratic prototype Hz2\mathcal{H}_{z^2} (Section 1.5 of the source) are already settled and do not count as resolutions.

References

  1. Dzimitry Dudko and Yusheng Luo. "Sierpinski carpet hyperbolic components of disjoint type are bounded." Preprint, arXiv:2509.25658, submitted 30 September 2025. arXiv identifier: arXiv:2509.25658. URL: https://arxiv.org/abs/2509.25658
  2. Dzimitry Dudko, Yusheng Luo, and Mikhail Lyubich. "Uniform a priori bounds for neutral renormalization. Variation II: ψ∙\psi^{\bullet}-ql Siegel maps." Preprint, arXiv:2509.23031, submitted 27 September 2025 (v2: 1 October 2025). arXiv identifier: arXiv:2509.23031. URL: https://arxiv.org/abs/2509.23031
  3. Curtis T. McMullen. "The classification of conformal dynamical systems." Current Developments in Mathematics, vol. 1995, pp. 323-360. International Press of Boston, 1995. DOI: 10.4310/cdm.1995.v1995.n1.a4. URL: https://doi.org/10.4310/cdm.1995.v1995.n1.a4