July  2011, 29(3): 929-952. doi: 10.3934/dcds.2011.29.929

On the topology of wandering Julia components

1. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, China

2. 

Département de Mathématiques, Université d'Angers, Angers, 49045, France

Received  December 2009 Revised  May 2010 Published  November 2010

It is known that for a rational map $f$ with a disconnected Julia set, the set of wandering Julia components is uncountable. We prove that all but countably many of them have a simple topology, namely having one or two complementary components. We show that the remaining countable subset $\Sigma$ is backward invariant. Conjecturally $\Sigma$ does not contain an infinite orbit. We give a very strong necessary condition for $\Sigma$ to contain an infinite orbit, thus proving the conjecture for many different cases. We provide also two sufficient conditions for a Julia component to be a point. Finally we construct several examples describing different topological structures of Julia components.
Citation: Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
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K. Pilgrim and L. Tan, Rational maps with disconnected Julia set,, Astérique, 261 (2000), 349. Google Scholar

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W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets,, Sci. China Ser. A, 52 (2009), 45. doi: 10.1007/s11425-008-0178-9. Google Scholar

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M. Shishikura, On the quasiconformal surgery of rational functions,, Ann. Sci. École Norm. Sup., 20 (1987), 1. Google Scholar

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D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains,, Ann. of Math., 122 (1985), 401. doi: 10.2307/1971308. Google Scholar

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Y. Zhai, A generalized version of Branner-Hubbard conjecture for rational functions,, Acta Mathematica Sinica, 26 (2010), 2199. doi: 10.1007/s10114-010-7632-7. Google Scholar

show all references

References:
[1]

A. F. Beardon, "Iteration of Rational Functions,", Springer-Verlag, (1991). Google Scholar

[2]

O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials,, Proc. London Math. Soc., 99 (2009), 275. doi: 10.1112/plms/pdn055. Google Scholar

[3]

C. T. McMullen, Automorphisms of rational maps,, in, (1988), 31. Google Scholar

[4]

J. Milnor, "Dynamics in One Complex Variable,", 3rd edition, (2006). Google Scholar

[5]

J. Milnor, On rational maps with two critical points,, Experimental Mathematics, 9 (2000), 481. Google Scholar

[6]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set,, Astérique, 261 (2000), 349. Google Scholar

[7]

K. Pilgrim and L. Tan, On disc-annulus surgery of rational maps,, in, (1999), 237. Google Scholar

[8]

W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets,, Sci. China Ser. A, 52 (2009), 45. doi: 10.1007/s11425-008-0178-9. Google Scholar

[9]

M. Shishikura, On the quasiconformal surgery of rational functions,, Ann. Sci. École Norm. Sup., 20 (1987), 1. Google Scholar

[10]

D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains,, Ann. of Math., 122 (1985), 401. doi: 10.2307/1971308. Google Scholar

[11]

Y. Zhai, A generalized version of Branner-Hubbard conjecture for rational functions,, Acta Mathematica Sinica, 26 (2010), 2199. doi: 10.1007/s10114-010-7632-7. Google Scholar

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