For the McMullen family of rational maps fλ(z)=zm+λ/znf_\lambda(z)=z^{m}+\lambda/z^{n} with integers m≥2m\ge 2, n≥1n\ge 1, let H0\mathcal{H}_0 denote the unique unbounded hyperbolic component of the parameter plane (the Cantor locus). He, Xiao and Yang proved that when m=n≥3m=n\ge 3, the Julia set J(fλ)J(f_\lambda) is a generalized Sierpiński gasket for every boundary parameter λ∈∂H0\lambda\in\partial\mathcal{H}_0: all Fatou components are Jordan domains, pairwise boundary intersections are uniformly bounded in size, no three component boundaries share a point, and the contact graph is connected. They explicitly conjecture the same conclusion for the asymmetric case m≠nm\neq n but their proof depends on structural results (Qiu–Wang–Yin; Qiu–Roesch–Wang–Yin) available only for the symmetric subfamily. The open problem is to prove or refute that for every exponent pair with m≠nm\neq n and every λ∈∂H0\lambda\in\partial\mathcal{H}_0, the Julia set of fλf_\lambda is a generalized Sierpiński gasket.

Background

For a rational map ff on the Riemann sphere, the Fatou set is the open set where the iterates fkf^{k} form a normal family, and the Julia set J(f)J(f) is its complement — the locus of chaotic dynamics. A central theme in one-dimensional complex dynamics is the geometry of Julia sets of singularly perturbed power maps, the simplest being the McMullen family 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\}, first studied by McMullen in 1988. A hyperbolic component of the parameter plane is a connected open set of parameters for which the map has a persistent attracting cycle; for this family there is a unique unbounded such component H0\mathcal{H}_0, called the Cantor locus because J(fλ)J(f_\lambda) is a Cantor set for every λ∈H0\lambda\in\mathcal{H}_0 (Devaney–Look–Uminsky). When λ\lambda lies on the boundary ∂H0\partial\mathcal{H}_0, the Julia set is connected (Devaney–Russell), and its fine structure becomes the central question. Besides Cantor sets, this family produces two classical fractal types of Julia sets: Sierpiński carpets (connected sets whose complementary domains are bounded by pairwise disjoint Jordan curves) and Sierpiński gaskets. The first Julia set homeomorphic to the classical Sierpiński gasket was found by Ushiki in the subfamily z2+λ/zz^{2}+\lambda/z at λ=−16/27\lambda=-16/27. Because essentially no other postcritically finite example (a map all of whose critical points have finite forward orbits) of the classical gasket type is known, Devaney, Moreno Rocha and Siegmund introduced generalized Sierpiński gaskets and proved that whenever fλf_\lambda is of Misiurewicz–Sierpiński type — meaning every free critical point (a critical point other than 00 and ∞\infty) is strictly preperiodic and lies on the boundary of the immediate basin of infinity (the Fatou component containing ∞\infty) — the Julia set J(fλ)J(f_\lambda) is a generalized Sierpiński gasket; this holds for all exponent pairs (m,n)(m,n). Following He–Xiao–Yang, a generalized Sierpiński gasket is a locally connected continuum (a compact connected set in which every point admits arbitrarily small connected neighborhoods) K⊂C^K\subset\widehat{\mathbb{C}} with empty interior such that: (1) every complementary component is a Jordan domain (a disk bounded by a simple closed curve); (2) there is a uniform NN such that the boundaries of any two complementary components meet in at most NN points; (3) no point lies on the boundary of three distinct complementary components; and (4) the contact graph — one vertex per complementary component, an edge when two boundaries touch — is connected. This notion generalizes both the recursively constructed gaskets of Devaney–Moreno Rocha–Siegmund and the tangential gaskets studied in quasiconformal geometry, where gasket Julia sets have become objects of a uniformization program (Luo–Ntalampekos; Luo–Zhang; Lodge–Lyubich–Merenkov–Mukherjee). Misiurewicz–Sierpiński parameters form only a countable subset of ∂H0\partial\mathcal{H}_0, which also contains parabolic parameters (where a critical orbit is attracted to a periodic cycle whose multiplier is a root of unity) and critically infinite parameters that are not postcritically finite (such as renormalizable parameters, around whose critical points small-scale copies of the original dynamics, i.e. polynomial-like return maps, recur), so the natural sharp question is whether every boundary parameter gives a gasket. For the symmetric subfamily zn+λ/znz^{n}+\lambda/z^{n} with n≥3n\ge 3, the deep parameter-plane analyses of Qiu–Wang–Yin and Qiu–Roesch–Wang–Yin (covering renormalization, boundary regularity of escape components, and the fact that Misiurewicz–Sierpiński parameters lie on ∂H0\partial\mathcal{H}_0) enabled He–Xiao–Yang to prove exactly this: if λ∈∂H0\lambda\in\partial\mathcal{H}_0 then J(zn+λ/zn)J(z^{n}+\lambda/z^{n}) is a generalized Sierpiński gasket for every n≥3n\ge 3. The Misiurewicz–Sierpiński subcase was previously covered by Devaney–Moreno Rocha–Siegmund for all (m,n)(m,n), but for m≠nm\neq n nothing is known about the remaining (parabolic, critically infinite, and other) boundary parameters, because the entire symmetric-case machinery has no asymmetric counterpart: in the symmetric family the 2n2n free critical points have only two critical values ±2λ\pm 2\sqrt{\lambda} and hence a single critical orbit (up to sign), whereas for m≠nm\neq n the m+nm+n free critical points take (m+n)/gcd⁡(m,n)(m+n)/\gcd(m,n) distinct critical values and thus split into several independent critical orbits, and the fine structure of ∂H0\partial\mathcal{H}_0 for unequal exponents has not been worked out (Steinmetz's general-m,nm,n study addresses the escape trichotomy and connectivity, not boundary contact geometry).

Problem Statement

For the McMullen family fλ(z)=zm+λ/znf_\lambda(z)=z^{m}+\lambda/z^{n} with integers m≥2m\ge 2, n≥1n\ge 1, m≠nm\neq n, let H0=H0(m,n)⊂C∖{0}\mathcal{H}_0=\mathcal{H}_0(m,n)\subset\mathbb{C}\setminus\{0\} be the unique unbounded hyperbolic component of the parameter plane, i.e. the unique unbounded connected open set of parameters on which fλf_\lambda has a persistent attracting cycle; this component is called the Cantor locus, and for every λ∈H0\lambda\in\mathcal{H}_0 the Julia set J(fλ)J(f_\lambda) is a Cantor set. Prove or refute the conjecture of He–Xiao–Yang: for every such asymmetric exponent pair and every boundary parameter λ∈∂H0\lambda\in\partial\mathcal{H}_0, the Julia set J(fλ)J(f_\lambda) is a generalized Sierpiński gasket — i.e. J(fλ)J(f_\lambda) is a locally connected continuum (a compact connected set in which every point admits arbitrarily small connected neighborhoods) with empty interior such that (1) every complementary component (Fatou component) is a Jordan domain, i.e. an open topological disk bounded by a simple closed curve; (2) there exists an integer 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 — the graph with one vertex per complementary component and an edge joining two vertices whose component boundaries meet — is connected.

References

  1. Xiaole He, Yingqing Xiao, Fei Yang, Rational maps whose Julia sets are generalized Sierpiński gaskets, arXiv:2503.18264 (v2, 2025), to appear in Indiana University Mathematics Journal; DOI 10.48550/arXiv.2503.18264, https://arxiv.org/abs/2503.18264
  2. Robert L. Devaney, Mónica Moreno Rocha, Stefan Siegmund, Rational maps with generalized Sierpinski gasket Julia sets, Topology and its Applications 154 (2007), no. 1, 11–27; DOI 10.1016/j.topol.2006.03.024, https://doi.org/10.1016/j.topol.2006.03.024
  3. Weiyuan Qiu, Xiaoguang Wang, Yongcheng Yin, Dynamics of McMullen maps, Advances in Mathematics 229 (2012), no. 4, 2525–2577; DOI 10.1016/j.aim.2011.12.026, https://doi.org/10.1016/j.aim.2011.12.026
  4. Weiyuan Qiu, Pascale Roesch, Xiaoguang Wang, Yongcheng Yin, Hyperbolic components of McMullen maps, Annales Scientifiques de l'École Normale Supérieure (4) 48 (2015), no. 3, 703–737; DOI 10.24033/asens.2256, https://doi.org/10.24033/asens.2256
  5. Robert L. Devaney, Daniel M. Look, David Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana University Mathematics Journal 54 (2005), no. 6, 1621–1634; DOI 10.1512/iumj.2005.54.2615, https://doi.org/10.1512/iumj.2005.54.2615
  6. Robert L. Devaney, Elizabeth D. Russell, Connectivity of Julia sets for singularly perturbed rational maps, in: Chaos, CNN, Memristors and Beyond, World Scientific, 2013, pp. 239–245; DOI 10.1142/9789814434805_0018, https://doi.org/10.1142/9789814434805_0018
  7. Norbert Steinmetz, On the dynamics of the McMullen family R(z)=zm+λ/zℓR(z)=z^{m}+\lambda/z^{\ell}, Conformal Geometry and Dynamics 10 (2006), 159–183; DOI 10.1090/s1088-4173-06-00149-4, https://doi.org/10.1090/s1088-4173-06-00149-4
  8. Weiyuan Qiu, Pascale Roesch, Yueyang Wang, Escape components of McMullen maps, Ergodic Theory and Dynamical Systems 43 (2023), no. 11, 3745–3775; DOI 10.1017/etds.2022.84, https://doi.org/10.1017/etds.2022.84
  9. Curtis T. McMullen, Automorphisms of rational maps, in: Holomorphic Functions and Moduli I (Berkeley, CA, 1986), Mathematical Sciences Research Institute Publications 10, Springer, New York, 1988, pp. 31–60; DOI 10.1007/978-1-4613-9602-4_3, https://doi.org/10.1007/978-1-4613-9602-4_3
  10. Sumio Ushiki, Julia sets with polyhedral symmetries, in: K. Shiraiwa (ed.), Dynamical Systems and Related Topics (Nagoya, 1990), Advanced Series in Dynamical Systems 9, World Scientific, River Edge, NJ, 1991, pp. 515–538; book DOI 10.1142/9789814539296, https://doi.org/10.1142/9789814539296
  11. Yusheng Luo, Dimitrios Ntalampekos, Uniformization of gasket Julia sets, arXiv:2411.17227 (2024); DOI 10.48550/arXiv.2411.17227, https://arxiv.org/abs/2411.17227
  12. Yusheng Luo, Yongquan Zhang, On quasiconformal non-equivalence of gasket Julia sets and limit sets, Ergodic Theory and Dynamical Systems 45 (2025), no. 11, 3465–3489; DOI 10.1017/etds.2025.10191, https://doi.org/10.1017/etds.2025.10191
  13. Russell Lodge, Mikhail Lyubich, Sergei Merenkov, Sabyasachi Mukherjee, On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections, Conformal Geometry and Dynamics 27 (2023), 1–54; DOI 10.1090/ecgd/379, https://doi.org/10.1090/ecgd/379
  14. Xiaole He, Yingqing Xiao, Fei Yang, Rational maps with Cantor bubble Julia sets, arXiv:2604.20096 (2026); DOI 10.48550/arXiv.2604.20096, https://arxiv.org/abs/2604.20096
  15. Yan Gao, Luxian Yang, Jinsong Zeng, Rigidity of McMullen Julia sets, arXiv:2606.24520 (2026); DOI 10.48550/arXiv.2606.24520, https://arxiv.org/abs/2606.24520