On the topology of wandering Julia components

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

On the topology of wandering Julia components

Guizhen Cui ^1, , Wenjuan Peng ^1, and Lei Tan ^2,

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

Abstract
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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.

Keywords: topology of wandering Julia components., Holomorphic dynamics, iteration of rational maps.

Mathematics Subject Classification: Primary: 37F10, 37F2.

Citation: Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929

References:

[1]	A. F. Beardon, "Iteration of Rational Functions," Springer-Verlag, New York, 1991.
[2]	O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. London Math. Soc., 99 (2009), 275-296. doi: 10.1112/plms/pdn055.
[3]	C. T. McMullen, Automorphisms of rational maps, in "Holomorphic Functions and Moduli I" (eds. D. Drasin, C. J. Earle, F. W. Gehring, I. Kra and A. Marden), Springer, (1988), 31-60.
[4]	J. Milnor, "Dynamics in One Complex Variable," 3rd edition, Princeton University Press, 2006.
[5]	J. Milnor, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.
[6]	K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérique, 261 (2000), 349-384.
[7]	K. Pilgrim and L. Tan, On disc-annulus surgery of rational maps, in "Proceedings of the International Conference in Dynamical Systems in Honor of Professor Liao Shan-tao 1998" (eds. Y. Jiang and L. Wen), World Scientific, (1999), 237-250.
[8]	W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Sci. China Ser. A, 52 (2009), 45-65. doi: 10.1007/s11425-008-0178-9.
[9]	M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29.
[10]	D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math., 122 (1985), 401-418. doi: 10.2307/1971308.
[11]	Y. Zhai, A generalized version of Branner-Hubbard conjecture for rational functions, Acta Mathematica Sinica, 26 (2010), 2199-2208. doi: 10.1007/s10114-010-7632-7.

show all references

References:

[1]	A. F. Beardon, "Iteration of Rational Functions," Springer-Verlag, New York, 1991.
[2]	O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. London Math. Soc., 99 (2009), 275-296. doi: 10.1112/plms/pdn055.
[3]	C. T. McMullen, Automorphisms of rational maps, in "Holomorphic Functions and Moduli I" (eds. D. Drasin, C. J. Earle, F. W. Gehring, I. Kra and A. Marden), Springer, (1988), 31-60.
[4]	J. Milnor, "Dynamics in One Complex Variable," 3rd edition, Princeton University Press, 2006.
[5]	J. Milnor, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.
[6]	K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérique, 261 (2000), 349-384.
[7]	K. Pilgrim and L. Tan, On disc-annulus surgery of rational maps, in "Proceedings of the International Conference in Dynamical Systems in Honor of Professor Liao Shan-tao 1998" (eds. Y. Jiang and L. Wen), World Scientific, (1999), 237-250.
[8]	W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Sci. China Ser. A, 52 (2009), 45-65. doi: 10.1007/s11425-008-0178-9.
[9]	M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29.
[10]	D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math., 122 (1985), 401-418. doi: 10.2307/1971308.
[11]	Y. Zhai, A generalized version of Branner-Hubbard conjecture for rational functions, Acta Mathematica Sinica, 26 (2010), 2199-2208. doi: 10.1007/s10114-010-7632-7.

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