Dicritical nilpotent holomorphic foliations
July 2018, 38(7): 3189-3221. doi: 10.3934/dcds.2018139
Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families
Jun Hu 1,2, , Oleg Muzician 3, and Yingqing Xiao 4,,
Department of Mathematics, Brooklyn College of CUNY, 2900 Bedford Avenue, Brooklyn, NY 11210, USA
Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, NY 10016, USA
Mathematics Department, BMCC of CUNY, 199 Chambers Street, New York, NY 10007, USA
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
* Corresponding author: Yingqing Xiao
Received February 2017 Revised November 2017 Published April 2018
Fund Project: The first author was supported by a Cycle 48 PSC-CUNY Research Award, and the third author was supported by the National Natural Science Foundation of China under grant Nos. 11301165, 11371126 and 11571099
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In [6], regularly ramified rational maps are constructed and Julia sets of these maps in some one-parameter families are explored through computer-generated pictures. It is observed that they have classifications similar to the Julia sets of maps in the families $ f_n^{c}(z) = z^n+\frac{c}{z^n}$, where $ n≥ 2$ and $ c$ is a complex number. A new type of Julia set is also presented, which has not appeared in the literature. We call such a Julia set an exploded McMullen necklace. We prove in this paper: if a map $ f$ in the one-parameter families given in [6] has a superattracting fixed point of order greater than 2, then its Julia set $ J(f)$ is either connected, a Cantor set, or a McMullen necklace (either exploded or not); if such a map $ f$ has a superattracting fixed point of order equal to 2, then $ J(f)$ is either connected or a Cantor set.
Keywords: Finite Kleinian groups, regularly ramified rational maps, Fatou components, and Julia sets.
Mathematics Subject Classification: Primary: 37F10, 37F45; Secondary: 30F40.
Citation: Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139
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Figure 1. Three Platonic solids.
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Figure 2. Four types of Julia sets for maps in the family $f_{(2, 4)}^{\lambda }$. In (a), a Cantor set with $\lambda = 2$; in (b), a non-escaping case of $v_{\lambda }$ with $\lambda = 3+3i$; in (c), a Sierpinski curve with $\lambda = 5$; and in (d), a McMullen necklace with $\lambda = 13$.
Figure 3. Three types of Julia sets for maps in the family $f_{(2, 2)}^{\lambda }$. In (a), a Cantor set with $\lambda = 1$; in (b), a non-escaping case of $v_{\lambda }$ with $\lambda = 3+5i$; in (c) and (d), Sierpinski curves with $\lambda = -4$ and $\lambda = 10$ respectively.
Figure 4. Three types of Julia sets for maps in the family $h_{(2, 4)}^{\lambda }$. In (a), $\lambda = 2$, a Cantor set; in (b), $\lambda = 3.467$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (c), $\lambda = 7$, a Sierpinski curve; in (d), $\lambda = -7$, a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Fatou set.
Figure 5. Four types of Julia sets for maps in the family $f_{(2, 3, 4)}^{\lambda }$. In (a), $\lambda = 20$, a Cantor set; in (b), $\lambda = 40+40i$, a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Fatou set; in (c), $\lambda = 500$, a Sierpinski curve; in (d) $\lambda = 1125$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (e), $\lambda = 1500$, an exploded McMullen necklace; (f) is a zoom of (e) in the middle.
Figure 6. Three types of Julia sets for maps in the family $h_{(2, 3, 4)}^{\lambda }$ (Note that $\infty $ is fixed). In (a), $\lambda = 1000$, a Cantor set; in (b), $\lambda = 890.5$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (c), $\lambda = 380i$, a non-escaping case of $v_{\lambda}$ with $v_{\lambda}$ in the Fatou set; in (d), $\lambda = 290$, a Sierpinski curve.
Figure 7. Here $B = B(0)$ and $T = B(\infty )$; the shadowed domain is an illustration for the domain $f_{\lambda }^{-1}(B(\infty ))$ proved in Lemma 3.16; $A_{in}$ and $A_{out}$ stand for the two annuli used in the proof of Proposition 3.20.
Figure 8. Four types of Julia sets for maps in the family $f_{(2, 3, 5)}^{\lambda }$. In (a), $\lambda = 200$, a Cantor set; in (b), $\lambda = 500$, a Sierpinski curve; in (c), $\lambda = 6000$, a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Fatou set; in (d), $\lambda = 20000$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (e), $\lambda = 30000$, an exploded McMullen necklace; (f) is a zoom of (e) in the middle.
Figure 9. Three types of Julia sets for maps in the family $h_{(2, 3, 5)}^{\lambda }$ (Note that $\infty $ is fixed). In (a), $\lambda = 15000-30000i$, a Cantor set; in (b), $\lambda = 12580-19760i$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (c), $\lambda = 9000+5000i$, a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Fatou set; in (d), $\lambda = 3500-6000i$, a Sierpinski curve.
Figure 10. Four types of Julia sets for maps in the family $f_{(2, 3, 3)}^{\lambda }$. In (a), $\lambda = 10$, a Cantor set; in (b), $\lambda = -200$, a Sierpinski curve; in (c), $\lambda = 30$, a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Fatou set; in (d), $\lambda = 290$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (e), $\lambda = 500$, an exploded McMullen necklace; (f) is a zoom of (e) in the middle. The black point in (a) or (f) stands for the origin, which is in the Fatou set.
Figure 11. Three types of Julia sets for maps in the family $h_{(2, 3, 3)}^{\lambda }$. In (a), $\lambda = 20i$, a Cantor set; in (b), $\lambda = 27.2899i$, an approximation of a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Julia set; in (c), $\lambda = 60$, a non-escaping case of $v_{\lambda }$ with $v_{\lambda }$ in the Fatou set; in (d), $\lambda = 120i$, a Sierpinski curve.
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Jun Hu Oleg Muzician Yingqing Xiao