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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 |
In [
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
![]()
References:
| [1] |
A. Beardon,
Iteration of Rational Functions, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
| [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. |
| [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. |
| [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. |
| [5] |
H. M. Farkas and I. Kra,
Riemann Surfaces, Springer-Verlag, 1980. |
| [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. |
| [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. |
| [8] |
J. Milnor,
Dynamics in one Complex Variable - Introductory Lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. |
| [9] |
——,
On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.
doi: 10.1080/10586458.2000.10504657. |
| [10] |
S. Morosawa,
Julia sets of sub-hyperbolic rational functions, Complex Variables Theory and Application, 41 (2000), 151-162.
doi: 10.1080/17476930008815244. |
| [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. |
| [12] |
G. T. Whyburn,
Topological characterization of the Sierpinski curve, Fund. Math., 45 (1958), 320-324.
doi: 10.4064/fm-45-1-320-324. |
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
A. Beardon,
Iteration of Rational Functions, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
| [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. |
| [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. |
| [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. |
| [5] |
H. M. Farkas and I. Kra,
Riemann Surfaces, Springer-Verlag, 1980. |
| [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. |
| [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. |
| [8] |
J. Milnor,
Dynamics in one Complex Variable - Introductory Lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. |
| [9] |
——,
On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.
doi: 10.1080/10586458.2000.10504657. |
| [10] |
S. Morosawa,
Julia sets of sub-hyperbolic rational functions, Complex Variables Theory and Application, 41 (2000), 151-162.
doi: 10.1080/17476930008815244. |
| [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. |
| [12] |
G. T. Whyburn,
Topological characterization of the Sierpinski curve, Fund. Math., 45 (1958), 320-324.
doi: 10.4064/fm-45-1-320-324. |
| [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. |
| [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. |
| [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. |
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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