Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families

Jun Hu; Oleg Muzician; Yingqing Xiao

doi:10.3934/dcds.2018139

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2018, Volume 38, Issue 7: 3189-3221. Doi: 10.3934/dcds.2018139

This issue Previous Article Next Article Dicritical nilpotent holomorphic foliations

Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families

1.
Department of Mathematics, Brooklyn College of CUNY, 2900 Bedford Avenue, Brooklyn, NY 11210, USA
2.
Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, NY 10016, USA
3.
Mathematics Department, BMCC of CUNY, 199 Chambers Street, New York, NY 10007, USA
4.
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

* Corresponding author: Yingqing Xiao
* Corresponding author: Yingqing Xiao

Received: February 2017

Revised: November 2017

Published: July 2018

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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Abstract

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.

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Mathematics Subject Classification: Primary: 37F10, 37F45; Secondary: 30F40.

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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 $\lambda = 13$.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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