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On the dynamics of generalized McMullen maps

Published online by Cambridge University Press:  05 June 2013

YINGQING XIAO ,
WEIYUAN QIU  and
YONGCHENG YIN
YINGQING XIAO
Affiliation:
College of Mathematics and Economics, Hunan University, Changsha, 410082, China email [email protected]
WEIYUAN QIU
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China email [email protected]
YONGCHENG YIN
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China email [email protected]
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Abstract

In this paper, we study the dynamics of the two-parameter family of rational maps

$$\begin{eqnarray*}{F}_{a, b} (z)= {z}^{n} + \frac{a}{{z}^{n} } + b.\end{eqnarray*}$$
We give the topological description of Julia sets and Fatou components of ${F}_{a, b} $ according to the dynamical behavior of the orbits of its free critical points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 2093 - 2112
Copyright
© Cambridge University Press, 2013 

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References

Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics, 132). Springer, New York, NY, 1991.Google Scholar
Blanchard, P., Devaney, R. L., Look, D. M., Seal, P. and Shapiro, Y.. Sierpiński curve Julia sets and singular perturbations of complex polynomials. Ergod. Th. & Dynam. Sys. 25 (2005), 1047–1055.Google Scholar
Blanchard, P., Devaney, R. L., Garijo, A., Marotta, S. and Russell, E. D.. Rabbits, basilicas, and other Julia sets wrapped in Sierpiński carpets. Complex Dynamics: Families and Friends. Eds. Schleicher, D. and Peters, A. K.. Wellesley, MA, 2009, pp. 277–296.CrossRefGoogle Scholar
Blanchard, P., Devaney, R. L., Garijo, A. and Russell, E. D.. A generalized version of the McMullen domain. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), 2309–2318.Google Scholar
Branner, B. and Hubbard, J. H.. The iteration of cubic polynomials. Acta Math. 169 (1992), 229–325.CrossRefGoogle Scholar
Cui, G.. Geometrically finite rational maps with given combinatorics. Preprint, 1997.Google Scholar
Devaney, R. L.. Structure of the McMullen domain in the parameter planes for rational maps. Fund. Math. 185 (2005), 267–285.Google Scholar
Kozma, R. T. and Devaney, R. L.. Julia sets converging to filled quadratic Julia sets. Ergod. Th. & Dynam. Sys. to appear; doi:10.1017/etds.2012.115.Google Scholar
Devaney, R. L., Look, D. M. and Uminsky, D.. The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54 (2005), 1621–1634.CrossRefGoogle Scholar
Devaney, R. L. and Look, D. M.. Symbolic dynamics for Sierpiński curve Julia set. J. Difference Equ. Appl. 11 (2005), 581–596.Google Scholar
Devaney, R. L. and Look, D. M.. A criterion for Sierpiński curve Julia sets. Topology Proc. 30 (2006), 1–17.Google Scholar
Devaney, R. L. and Marotta, S. M.. Evolution of the McMullen domain for singularly perturbed rational maps. Topology Proc. 32 (2008), 301–320.Google Scholar
Devaney, R. L., Rocha, M. M. and Siegmund, S.. Rational maps with generalized Sierpiński gasket Julia sets. Topology Appl. 154 (2007), 11–27.CrossRefGoogle Scholar
Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), 287–314.CrossRefGoogle Scholar
Jang, H. G. and Steinmetz, N.. On the dynamics of the rational family ${f}_{t} (z)= - \mathop{({z}^{2} - 2)}\nolimits ^{2} / 4({z}^{2} - 1)$. Comput. Methods Funct. Theory 12 (2012), 1–17.CrossRefGoogle Scholar
McMullen, C. T.. Automorphisms of rational maps. Holomorphic Function and Moduli, Vol. 1 (Mathematical Sciences Research Institute Publications, 10). Springer, New York, 1988.Google Scholar
Milnor, J.. Dynamics in One Complex Variable – Introductory Lectures, 2nd edn. Vieweg, Braunschweig, 1999.Google Scholar
Milnor, J.. Local connectivity of Julia sets: expository lectures. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Ed. Tan, Lei. Cambridge University Press, Cambridge, 2000, pp. 67–116.Google Scholar
Morosawa, S.. Julia sets of sub-hyperbolic rational functions. Complex Var. Theory Appl. 41 (2000), 151–162.Google Scholar
Qiu, W. and Yin, Y.. Proof of Branner-Hubbard conjecture on the Cantor Julia set. Sci. China Ser. A 52 (2009), 45–65.CrossRefGoogle Scholar
Qiu, W., Wang, X. and Yin, Y.. Dynamics of McMullen maps. Adv. Math. 229 (2012), 2525–2577.Google Scholar
Roesch, P.. On captures for the family ${f}_{\lambda } (z)= {z}^{2} + \lambda / {z}^{2} $. Dynamics on the Riemann Sphere. Eds. Hjorth, P. G. and Petersen, C. L.. European Mathematical Society, Zürich, 2006, pp. 121–130.CrossRefGoogle Scholar
Steinmetz, N.. On the dynamics of the McMullen family $R(z)= {z}^{n} + \lambda / {z}^{\ell } $. Conform. Geom. Dyn. 10 (2006), 159–183.Google Scholar
Steinmetz, N.. Sierpiński and non-Sierpiński curve Julia sests in families of rational maps. J. Lond. Math. Soc. (2). 78 (2008), 290–304.Google Scholar
Whyburn, G. T.. Topological characterization of the Sierpiński curve. Fund. Math. 45 (1958), 320–324.Google Scholar