# American Institute of Mathematical Sciences

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

 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

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.

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.

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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##### References:
 [1] A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6.  Google Scholar [2] R. L. Devaney, Dynamics of $z^n+λ /z^n$; Why is the case $n = 2$ crazy, Contemp. Math., 573 (2012), 49-65.  doi: 10.1090/conm/573/11379.  Google Scholar [3] R. L. Devaney, D. M. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana University Mathematics Journal, 54 (2005), 1621-1634.  doi: 10.1512/iumj.2005.54.2615.  Google Scholar [4] R. L. Devaney and E. D. Russell, Connectivity of Julia sets for singularly perturbed rational maps, in Chaos, CNN, Memristors and Beyond, World Scientific, (2013), 239--245. doi: 10.1142/9789814434805_0018.  Google Scholar [5] H. M. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, 1980.  Google Scholar [6] J. Hu, F. G. Jimenez and O. Muzician, Rational maps with half symmetries, Julia sets, and Multibrot sets in parameter planes, Contemp. Math., 573 (2012), 119-146.  doi: 10.1090/conm/573/11393.  Google Scholar [7] C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli, Vol. Ⅰ (Berkeley, CA, 1986), 31-60, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar [8] J. Milnor, Dynamics in one Complex Variable - Introductory Lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999.  Google Scholar [9] ——, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.  doi: 10.1080/10586458.2000.10504657.  Google Scholar [10] S. Morosawa, Julia sets of sub-hyperbolic rational functions, Complex Variables Theory and Application, 41 (2000), 151-162.  doi: 10.1080/17476930008815244.  Google Scholar [11] M. Stiemer, Rational maps with Fatou components of arbitrary connectivity number, Computational Methods and Function Theory, 7 (2007), 415-427.  doi: 10.1007/BF03321654.  Google Scholar [12] G. T. Whyburn, Topological characterization of the Sierpinski curve, Fund. Math., 45 (1958), 320-324.  doi: 10.4064/fm-45-1-320-324.  Google Scholar [13] Y. Xiao and W. Qiu, The rational maps $F_{λ }(z)=z^m+\frac{λ }{z^d}$ have no Herman rings, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 403-407.  doi: 10.1007/s12044-010-0044-x.  Google Scholar [14] Y. Xiao, W. Qiu and Y. Yin, On the dynamics of generalized McMullen maps, Ergod. Th. & Dynam. Sys., 34 (2014), 2093-2112.  doi: 10.1017/etds.2013.21.  Google Scholar [15] Y. Xiao and F. Yang, Singular perturbations of the unicritical polynomials with two parameters, Ergod. Th. & Dynam. Sys., 37 (2017), 1997-2016.  doi: 10.1017/etds.2015.114.  Google Scholar
Three Platonic solids.
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 $\lambda = 13$.
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.
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.
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.
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.
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.
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.
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.
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.
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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