A rational map is bicritical if it has exactly two critical points, each of local degree d. Milnor showed that the moduli space of degree-d bicritical rational maps is naturally Md≅C2\mathcal{M}_d \cong \mathbb{C}^2, and its hyperbolic components follow the quadratic A/B/C/D classification. Nie and Pilgrim proved that strict type D components (two distinct attracting cycles, each of period at least two) are bounded, and explicitly stated that they do not know whether type C components are bounded or unbounded, nor how to construct a degenerating family inside one. Type C components are the capture components: both critical points are attracted to one common attracting cycle, but only one lies in its immediate basin. This problem asks to decide, for every degree d≥2d \ge 2, whether every type C hyperbolic component of Md\mathcal{M}_d has compact closure, or whether some capture component is unbounded. It is a concrete remaining open case, within the bicritical family, of McMullen's conjecture that hyperbolic components with Sierpiński carpet Julia sets are bounded — a conjecture ranging over hyperbolic components of the full moduli space of degree-dd rational maps: the degree-two and bicritical strict type D cases are settled (Epstein; Nie–Pilgrim), quasi post-critically finite degenerations are bounded (Luo), and disjoint-type carpet components are bounded (Dudko–Luo), while the capture case is open in every degree.

Background

A rational map is a holomorphic self-map f ⁣:P1→P1f\colon \mathbb{P}^1 \to \mathbb{P}^1 of the Riemann sphere of degree d≥2d \ge 2. It 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 ratd\mathrm{rat}_d of holomorphic conjugacy classes of degree-dd rational maps, and a connected component HH of this subset is a hyperbolic component. A basic structural question, asked in particular by Epstein and by McMullen, is when a hyperbolic component is bounded, i.e. has compact closure in moduli space.

A rational map is bicritical if it has exactly two critical points, each of local degree dd; equivalently f=M∘zd∘Nf = M \circ z^d \circ N for Möbius transformations M,N∈Aut(P1)M, N \in \mathrm{Aut}(\mathbb{P}^1). Every quadratic rational map (d=2d = 2) is automatically bicritical. Milnor showed that the moduli space of bicritical degree-dd maps, i.e. the image Md⊂ratd\mathcal{M}_d \subset \mathrm{rat}_d, is biholomorphic to C2\mathbb{C}^2, with explicit coordinates given by two elementary symmetric functions of the multipliers (the derivative of the first return map at a periodic point) of the d+1d+1 fixed points; the inclusion Md↪ratd\mathcal{M}_d \hookrightarrow \mathrm{rat}_d is proper, so a component H⊂MdH \subset \mathcal{M}_d has compact closure in Md\mathcal{M}_d if and only if its image has compact closure in ratd\mathrm{rat}_d. Following the classification of hyperbolic components in the quadratic case, components of Md\mathcal{M}_d are of type A (both critical points lie in the immediate basin of an attracting fixed point), type B (both critical points in the immediate basin of a common attracting cycle of period at least two), type C — the capture components, where both critical points are attracted to a common attracting cycle but only one lies in the immediate basin — and type D, where each map has two distinct attracting cycles. Here the Fatou set is the largest open subset of the sphere on which the iterates of ff form a normal family, its connected components are Fatou components, and the immediate basin of an attracting cycle is the Fatou component containing that cycle. By results of Makienko, type A components and certain type B components are unbounded. Epstein proved boundedness for quadratic maps with two distinct attracting cycles of period at least two, and Nie–Pilgrim extended this to strict type D components of Md\mathcal{M}_d (neither attracting cycle a fixed point) in all degrees, using arithmetic methods: a degenerating family is transported to dynamics on the Berkovich projective line (a non-Archimedean analogue of the Riemann sphere with a richer point set) over a non-Archimedean field of Puiseux series, where rescaling limits (renormalized limits of iterates, in the sense of Kiwi's theory) are analyzed via Rivera-Letelier's arithmetic results and Epstein's refined Fatou–Shishikura inequality.

Two features make the type C case different. First, unlike type D components — which are algebraically parameterized by the two attracting multipliers, so that a degenerating family can be reduced to a holomorphic one-parameter family — components of types A, B, and C are, in the words of Nie–Pilgrim, transcendental objects, and for type C they state they do not even know how to extract a degenerating holomorphic family ftf_t with ftk∈Hf_{t_k} \in H. Second, capture components are the natural home of Sierpiński carpet Julia sets: a Sierpiński carpet Julia set means a Julia set homeomorphic to the classical Sierpiński carpet, i.e. a planar continuum obtained from the sphere by removing countably many disjoint Jordan domains whose union is dense and whose diameters tend to zero; a hyperbolic map whose Fatou components are all simply connected, with pairwise disjoint Jordan curve boundaries that are dense in the sphere and have diameters tending to zero, has a Sierpiński carpet Julia set, and this configuration is typical for capture dynamics. McMullen, motivated by Thurston's compactness theorem for acylindrical hyperbolic 3-manifolds, conjectured in the early 1990s that hyperbolic components with Sierpiński carpet Julia sets are bounded. A familiar one-parameter instance is the family of McMullen maps zn+λ/zdz^n + \lambda/z^d: its 'Sierpiński holes' are the loci where the free critical orbit is eventually captured by the basin of the superattracting fixed point at infinity, producing a Sierpiński carpet Julia set (the boundary geometry of these components is understood from work of Qiu–Roesch–Wang–Yin). After Epstein's degree-two theorem, Nie–Pilgrim's strict type D theorem, Luo's boundedness result for quasi post-critically finite degenerations (a tameness hypothesis on a degenerating family, under which the limits exist as geometrically finite maps), and Dudko–Luo's theorem for disjoint-type carpet components (every critical point in its own periodic Fatou cycle), the capture case is precisely the part of McMullen's conjecture that remains open within the bicritical family Md\mathcal{M}_d; the conjecture itself ranges over hyperbolic components of the full moduli space of degree-dd rational maps, so cases beyond the bicritical family also remain.

Problem Statement

Let d≥2d \ge 2 and let Md≅C2\mathcal{M}_d \cong \mathbb{C}^2 be the moduli space of bicritical rational maps of degree dd (holomorphic conjugacy classes of rational maps with exactly two critical points, each of local degree dd; for d=2d = 2 this is the full moduli space of quadratic rational maps). Call a class [f]∈Md[f] \in \mathcal{M}_d hyperbolic if every critical point of ff converges under iteration to an attracting periodic cycle; the hyperbolic classes form an open subset of Md\mathcal{M}_d, and a connected component HH of it is a hyperbolic component. Let HH be a hyperbolic component of type C, i.e. a capture component: for [f]∈H[f] \in H, both critical points of ff are attracted to one common attracting periodic cycle, but only one of them lies in the immediate basin of that cycle, the immediate basin being the connected component of the Fatou set (the maximal open set on which the iterates form a normal family) that contains the cycle. Determine, for every degree d≥2d \ge 2 and every type C hyperbolic component H⊂MdH \subset \mathcal{M}_d, whether HH is bounded — equivalently, whether HH has compact closure in Md\mathcal{M}_d, or equivalently whether the image of HH has compact closure in the full moduli space ratd\mathrm{rat}_d of degree-dd rational maps. The problem includes, as the capture case of McMullen's conjecture that hyperbolic components with Sierpiński carpet Julia sets are bounded (a Sierpiński carpet Julia set being defined in the Background above), the sub-question of whether type C components whose maps have Sierpiński carpet Julia sets are bounded; no finite-size, symmetry, or combinatorial restriction on the components is imposed. A secondary open question stated by the source, separate from and not required by the decision problem above, is how one could extract a degenerating holomorphic family inside an unbounded type C component: a positive resolution would make the extraction question vacuous, while an explicit unbounded component would not by itself produce such a family.

References

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