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Rational maps whose Julia sets are Cantor circles

Published online by Cambridge University Press:  19 August 2013

WEIYUAN QIU ,
FEI YANG  and
YONGCHENG YIN
WEIYUAN QIU
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai, 200433, PR China email wyqiu@fudan.edu.cn
FEI YANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, 210093, PR China email yangfei_math@163.com
YONGCHENG YIN
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, PR China email yin@zju.edu.cn
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Abstract

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 499 - 529
Copyright
© Cambridge University Press, 2013 

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References

Beardon, A.. Iteration of Rational Functions (Graduate Texts in Mathematics, 132). Springer, New York, 1991.Google Scholar
Cui, G.. Dynamics of rational maps, topology, deformation and bifurcation. Preprint, May 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997).Google Scholar
Devaney, R., Look, D. and Uminsky, D.. The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54 (6) (2005), 1621–1634.Google Scholar
Haïssinsky, P. and Pilgrim, K.. Quasisymmetrically inequivalent hyperbolic Julia sets. Rev. Mat. Iberoam. 28 (4) (2012), 1025–1034.CrossRefGoogle Scholar
McMullen, C.. Automorphisms of rational maps. Holomorphic Functions and Moduli I (Mathematical Sciences Research Institute Publications, 10). Springer, New York, 1988.Google Scholar
Pilgrim, K. and Tan, L.. Rational maps with disconnected Julia sets. Astérisque 261 (2000), 349–383.Google Scholar
Qiu, W., Wang, X. and Yin, Y.. Dynamics of McMullen maps. Adv. Math. 229 (4) (2012), 2525–2577.Google Scholar
Qiu, W. and Yin, Y.. Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A 52 (1) (2009), 45–65.Google Scholar
Steinmetz, N.. On the dynamics of the McMullen family $R(z)= {z}^{m} + \lambda / {z}^{\ell } $. Conform. Geom. Dyn. 10 (2006), 159–183.Google Scholar
Tan, L. and Yin, Y.. Local connectivity of the Julia sets for geometrically finite rational maps. Sci. China Ser. A 39 (1) (1996), 39–47.Google Scholar