For the McMullen family of rational maps fλ(z)=zm+λ/znf_\lambda(z)=z^m+\lambda/z^n (m≥2m\ge 2, n≥1n\ge 1, λ∈C∖{0}\lambda\in\mathbb{C}\setminus\{0\}), He–Xiao–Yang proved that when m=n≥3m=n\ge 3 and λ\lambda lies on the boundary of the unique unbounded hyperbolic component H0H_0 (the Cantor locus), the Julia set J(fλ)J(f_\lambda) is a generalized Sierpiński gasket, and they explicitly conjectured the same conclusion for the asymmetric case m≠nm\neq n. This problem asks to settle that conjecture: either prove that for every pair of integers m≥2m\ge 2, n≥1n\ge 1 with m≠nm\neq n and every λ∈∂H0\lambda\in\partial H_0 the Julia set J(fλ)J(f_\lambda) satisfies the four defining gasket properties (complementary components are Jordan domains, pairwise boundary intersections uniformly bounded, no triple contact point, connected contact graph), or exhibit an asymmetric pair (m,n)(m,n) and a parameter λ∗∈∂H0\lambda^*\in\partial H_0 for which at least one property fails.

Background

A rational map on the Riemann sphere splits the sphere into the Fatou set, where the iterates form a normal family (every sequence of iterates has a locally uniformly convergent subsequence), and its complement the Julia set, the locus of chaotic dynamics. The McMullen family is the one-parameter family of rational maps

fλ(z)=zm+λ/zn,m≥2, n≥1, λ∈C∖{0},f_\lambda(z)=z^m+\lambda/z^n,\qquad m\ge 2,\ n\ge 1,\ \lambda\in\mathbb{C}\setminus\{0\},

a singular perturbation of the polynomial zmz^m (the term λ/zn\lambda/z^n inserts a pole at the origin) introduced by McMullen (1988), who showed that for 1/m+1/n<11/m+1/n<1 and small ∣λ∣|\lambda| the Julia set is a Cantor set of circles. The point ∞\infty is a superattracting fixed point (an attracting fixed point at which the derivative vanishes); its immediate basin (the Fatou component containing ∞\infty) is denoted BλB_\lambda. Besides 00 and ∞\infty, the map has m+nm+n free critical points (the further zeros of the derivative), the solutions of zm+n=nλ/mz^{m+n}=n\lambda/m. Devaney–Look–Uminsky (2005) established an escape trichotomy: when the free critical points lie in the basin of ∞\infty, the Julia set is a Cantor set, a Cantor set of circles, or a Sierpiński curve (a planar set homeomorphic to the standard Sierpiński carpet) depending on the escape speed. Steinmetz (2006) developed the dynamics of the general (asymmetric) family.

A parameter is hyperbolic if all critical orbits converge to attracting cycles; a hyperbolic component is a connected component of the set of hyperbolic parameters. For every m≥2m\ge 2, n≥1n\ge 1 the λ\lambda-plane of this family contains a unique unbounded hyperbolic component H0H_0, called the Cantor locus: when λ∈H0\lambda\in H_0 the Julia set J(fλ)J(f_\lambda) is a Cantor set (Devaney–Look–Uminsky 2005), and by Devaney–Russell (2013) J(fλ)J(f_\lambda) is connected for every λ∈∂H0\lambda\in\partial H_0. A Misiurewicz–Sierpiński parameter is one at which every free critical point is strictly preperiodic (eventually periodic but not periodic) and lands on ∂Bλ\partial B_\lambda; Devaney–Rocha–Siegmund (2007) proved that the Julia set of such a parameter is a generalized Sierpiński gasket. For the symmetric subfamily zn+λ/znz^n+\lambda/z^n (n≥3n\ge 3), Qiu–Wang–Yin (2012) and Qiu–Roesch–Wang–Yin (2015) developed a detailed theory of the parameter and dynamical planes: ∂Bλ\partial B_\lambda is a Jordan curve (a simple closed curve) off H0H_0; λ∈∂H0\lambda\in\partial H_0 if and only if ∂Bλ\partial B_\lambda contains either the whole free critical set or a parabolic cycle (a periodic cycle whose multiplier is a root of unity); and, absent Siegel disks (Fatou components on which an iterate is conjugate to an irrational rotation) and renormalization, every Fatou component is a Jordan domain (a domain bounded by a Jordan curve) and the Julia set is locally connected. Here a renormalization is a polynomial-like restriction of an iterate (a proper holomorphic self-map of a Jordan disk pair hybrid equivalent to a polynomial), and these results classify exactly when such restrictions occur on ∂H0\partial H_0.

He–Xiao–Yang (2025, to appear in Indiana Univ. Math. J.) introduced the following definition: a generalized Sierpiński gasket is a locally connected continuum (a compact connected set) K⊂C^K\subset\widehat{\mathbb{C}} with empty interior such that (1) each complementary component of KK is a Jordan domain; (2) there exists N≥1N\ge 1 such that the boundaries of any two complementary components of KK share at most NN points; (3) no three complementary components have a common boundary point; and (4) the contact graph of KK (one vertex per complementary component, an edge when two touch) is connected. Their Theorem 1.1 states: if λ∈∂H0\lambda\in\partial H_0, then the Julia set of fλ(z)=zm+λ/znf_\lambda(z)=z^m+\lambda/z^n with m=n≥3m=n\ge 3 is a generalized Sierpiński gasket. This extends the Misiurewicz–Sierpiński result of Devaney–Rocha–Siegmund to the whole boundary of the Cantor locus, including critically infinite and parabolic parameters. The proof, however, consumes the symmetric-only theory of Qiu–Wang–Yin and Qiu–Roesch–Wang–Yin at every step. The obstruction is structural: in the symmetric case the free critical points share only the two critical values (images of critical points) ±2λ\pm 2\sqrt{\lambda}, so all free critical orbits have the same fate; for m≠nm\neq n a direct computation shows there are (m+n)/gcd⁡(m,n)(m+n)/\gcd(m,n) distinct critical values, so free critical orbits can decouple and the all-or-nothing critical behavior underlying the symmetric arguments is lost. Gasket Julia sets are also of independent current interest: they are the subject of an active program in quasisymmetric geometry (the metric study of fractal sets under deformations stricter than homeomorphisms) and uniformization (Luo–Ntalampekos 2024; Luo–Zhang 2025) and of constructions via Kleinian groups (discrete groups of Möbius transformations of the sphere) and Schwarz reflections (iterated anti-holomorphic reflections across analytic arcs) (Lodge–Lyubich–Merenkov–Mukherjee 2023), which makes the supply of gasket examples beyond geometrically finite maps (maps whose critical orbits are all eventually periodic or converge to attracting or parabolic cycles) a recognized gap.

Problem Statement

For integers m≥2m\ge 2 and n≥1n\ge 1 with m≠nm\neq n, consider the McMullen family of rational maps fλ(z)=zm+λ/znf_\lambda(z)=z^m+\lambda/z^n on the Riemann sphere, with parameter λ∈C∖{0}\lambda\in\mathbb{C}\setminus\{0\}. Call a parameter hyperbolic if every critical orbit of fλf_\lambda converges to an attracting cycle; the hyperbolic components are the connected components of the set of hyperbolic parameters in the λ\lambda-plane, and for every such pair (m,n)(m,n) that plane contains a unique unbounded hyperbolic component H0H_0, the Cantor locus (so named because J(fλ)J(f_\lambda) is a Cantor set for λ∈H0\lambda\in H_0). Let J(fλ)J(f_\lambda) denote the Julia set of fλf_\lambda, the complement of the Fatou set (the largest open set on which the iterates of fλf_\lambda form a normal family, i.e. every sequence of iterates has a locally uniformly convergent subsequence). Determine whether the following statement, conjectured by He–Xiao–Yang immediately after their Theorem 1.1 (which proves the case m=n≥3m=n\ge 3; the conjecture sentence reads "We guess that the result in Theorem 1.1 is also true for m≠nm\neq n"), is true: for every λ∈∂H0\lambda\in\partial H_0, the Julia set J(fλ)J(f_\lambda) is a generalized Sierpiński gasket, i.e. a locally connected continuum (a compact connected set) with empty interior such that (1) each complementary component of J(fλ)J(f_\lambda) is a Jordan domain (a domain whose boundary is a Jordan curve, i.e. a simple closed curve); (2) there exists N≥1N\ge 1 such that the boundaries of any two complementary components share at most NN points; (3) no three complementary components have a common boundary point; and (4) the contact graph of the complementary components (one vertex per component, an edge when two touch) is connected. A complete answer either proves this for all pairs (m,n)(m,n) with m≥2m\ge 2, n≥1n\ge 1, m≠nm\neq n, or exhibits a specific pair (m,n)(m,n) and a specific parameter λ∗∈∂H0\lambda^*\in\partial H_0 together with a proof that at least one of the four properties fails for J(fλ∗)J(f_{\lambda^*}).

References

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