For a bicritical rational map (exactly two critical points), the moduli space of conjugacy classes is biholomorphic to C2\mathbb{C}^2. Nie and Pilgrim proved that hyperbolic components of strict type D (two distinct attracting cycles of period at least two) are bounded in this moduli space, and type A and some type B components are known to be unbounded. The status of type C components — capture components in which both critical points are attracted to a single attracting cycle but only one lies in its immediate basin — is left completely open: it is not known whether such a component can be unbounded, nor whether all are bounded, nor even how to extract a degenerating holomorphic one-parameter family inside an unbounded one. The problem is to settle this boundedness question.

Background

A rational map f:P1→P1f:\mathbb{P}^1\to\mathbb{P}^1 of degree d≥2d\ge 2 is a dynamical system on the Riemann sphere. It is called hyperbolic if every critical point of ff converges under iteration to an attracting periodic cycle; hyperbolic maps form an open, conjecturally dense subset of the space of all rational maps, and a connected component HH of this subset is a hyperbolic component. Since Möbius conjugation f↦M∘f∘M−1f\mapsto M\circ f\circ M^{-1} preserves iteration, one works in the moduli space ratd\mathrm{rat}_d of holomorphic conjugacy classes of degree-dd rational maps, a complex affine variety of dimension 2d−22d-2.

A rational map is bicritical if it has exactly two distinct critical points (each then of multiplicity d−1d-1); equivalently f=M∘zd∘Nf=M\circ z^d\circ N for Möbius maps M,NM,N. Milnor showed that the moduli space Md\mathcal{M}_d of bicritical rational maps is biholomorphic to C2\mathbb{C}^2, with explicit coordinates given by two elementary symmetric functions (σ1,σd)(\sigma_1,\sigma_d) of the multipliers of the d+1d+1 fixed points, and that the natural inclusion Md↪ratd\mathcal{M}_d\hookrightarrow\mathrm{rat}_d is proper, so a component H⊂MdH\subset\mathcal{M}_d is bounded in Md\mathcal{M}_d if and only if its image is bounded in ratd\mathrm{rat}_d. This family includes the unicritical polynomial families zd+cz^d+c (for d=2d=2 it is the space of all quadratic rational maps).

Hyperbolic components in Md\mathcal{M}_d are classified as in the quadratic rational case studied by Milnor: type A — the map has an attracting fixed point; type B — there is an attracting cycle of period at least two whose immediate basin (the union of the periodic Fatou components, i.e. the periodic connected components of the open set where iterates form a normal family) contains both critical points; type C, a capture component — both critical points are attracted to a single attracting periodic cycle but only one lies in the immediate basin; and type D — the map has two distinct attracting cycles (strict type D when neither is a fixed point).

The guiding question is when a hyperbolic component has compact closure in moduli space. Makienko, and later Tan via pinching deformations (continuous paths in parameter space obtained by collapsing suitable simple closed curves in the Fatou set to points), gave general sufficient conditions for unboundedness which imply that type A components and certain type B components are unbounded. In the other direction, Epstein proved that strict type D components of quadratic rational maps are bounded, and Nie–Pilgrim extended this to strict type D components of Md\mathcal{M}_d for all degrees. Their proof is by contradiction: a degenerating sequence in HH is converted, using the multiplier coordinates of the two attracting cycles (which make strict type D components semi-algebraic, in fact algebraically equivalent to a product of disks), into a holomorphic one-parameter family ftf_t degenerating as t→0t\to 0; a rescaling limit (a nonconstant limit g=lim⁡kMk∘fkqk∘Mk−1g=\lim_k M_k\circ f_k^{q_k}\circ M_k^{-1} of conjugated iterates under Möbius changes of coordinates) is extracted and analyzed as a dynamical system on the Berkovich projective line — a non-Archimedean (valuation-theoretic) compactification of P1\mathbb{P}^1 over a field of Laurent series — and an over-determined set of constraints on the critical dynamics of the limit, combining arithmetic results of Rivera-Letelier with Epstein's refinement of the Fatou–Shishikura inequality (which bounds the number of non-repelling cycles of a rational map by its number of critical points), yields a contradiction. Kabelka's normal-form analysis of bicritical maps supplies the nonlinear rescaling limit. The case of type C components is left open by that machinery, and the authors stress that unlike type D, components of types A, B, and C are transcendental objects: no algebraic parametrization of them is known, and if a type C component were unbounded, they do not even know how to extract a degenerating holomorphic one-parameter family ftf_t remaining in the component along a sequence tk→0t_k\to 0.

The question sits inside a broader program. A compact plane set is a Sierpiński carpet if it is homeomorphic to the standard middle-thirds carpet; a hyperbolic map with connected Julia set and at least three Fatou components that are Jordan domains with pairwise disjoint closures has a Sierpiński carpet Julia set. McMullen conjectured in the 1990s, by analogy with Thurston's compactness theorem for acylindrical hyperbolic 3-manifolds, that every Sierpiński carpet hyperbolic component is bounded in the moduli space of rational maps; capture components are among the principal expected instances. Dudko and Luo (2025) proved this conjecture for components of disjoint type (in which every critical point lies in its own periodic Fatou component, giving 2d−22d-2 attracting cycles) and record that the general conjecture remains wide open beyond degree 2. Type C components, having a single attracting cycle, are not of disjoint type.

Problem Statement

Determine the boundedness status of capture hyperbolic components in the moduli space of bicritical rational maps. Precisely: for each degree d≥2d\ge 2, let Md≅C2\mathcal{M}_d\cong\mathbb{C}^2 be the moduli space of bicritical rational maps of degree dd, and let H⊂MdH\subset\mathcal{M}_d be a hyperbolic component of type C, i.e. every map in HH has a single attracting periodic cycle, of period at least two, such that both critical points lie in its full basin (the set of all points whose orbits converge to the cycle) while exactly one lies in the immediate basin (the periodic Fatou components of that cycle, as opposed to their iterated preimages). Decide which of the following holds:

(a) every type C component HH has compact closure in Md\mathcal{M}_d, with a proof that rules out degenerating sequences in HH (thereby completing the boundedness classification of hyperbolic components of Md\mathcal{M}_d begun by Epstein and by Nie–Pilgrim for strict type D, and by Makienko and Tan for the unbounded type A and certain type B cases); or

(b) there exists a degree dd and an unbounded type C component H⊂MdH\subset\mathcal{M}_d, exhibited together with a rigorous proof of unboundedness (for instance via an explicit curve or sequence in HH whose coordinates in Md\mathcal{M}_d tend to infinity).

Nie and Pilgrim record one auxiliary open difficulty alongside the boundedness question: if a type C component HH were unbounded, they do not know how to extract a degenerating holomorphic one-parameter family ftf_t, tt in the unit disk, such that ftk∈Hf_{t_k}\in H for some sequence tk→0t_k\to 0 — the step that in the type D argument is supplied by semi-algebraicity. This extraction problem is recorded here as context for the difficulty of the question rather than as an additional requirement on a resolution: settling either branch (a) or (b) serves the one scientific objective, the parameter-space geometry of capture components of bicritical maps. The authors' secondary method-interpretation questions (extending Epstein's analytic arguments to bicritical degree d>2d>2, and reinterpreting those arguments in arithmetic/Berkovich terms) are likewise recorded as background context, not as part of the problem.

References

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