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On the connectivity of the Julia set of a finitely generated rational semigroup

Author(s): Yeshun Sun; Chung-Chun Yang
Journal: Proc. Amer. Math. Soc. 130 (2002), 49-52.
MSC (2000): Primary 37F10, 37F50
Posted: May 3, 2001
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Abstract:

In this paper we show that the Julia set $J(G)$ of a finitely generated rational semigroup $G$ is connected if the union of the Julia sets of generators is contained in a subcontinuum of $J(G)$. Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

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A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc. 73(1996), 358-384. MR 97e:58198

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J. Milnor, Dynamics in one complex variable: Introductory lectures, Vieweg Verlag, 1999. CMP 2000:03

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S. B. Nadler, Continuum Theory: An introduction, Marcel Dekker, 1992. MR 93m:54002

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H. Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ. 37(1997), 717-733. MR 99h:30028

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W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Sci. Bull. 37(1992), 969-971.

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Additional Information:

Yeshun Sun
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
Address at time of publication: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, Peoples' Republic of China
Email: maysun@ust.hk, sun@math.zju.edu.cn

Chung-Chun Yang
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
Email: mayang@ust.hk

DOI: 10.1090/S0002-9939-01-06097-X
PII: S 0002-9939(01)06097-X
Keywords: Connectivity, Julia set, rational semigroup
Received by editor(s): May 4, 2000
Posted: May 3, 2001
Additional Notes: This research was partially supported by a UGC grant of Hong Kong, Project No. 6070/98P
Communicated by: Linda Keen
Copyright of article: Copyright 2001, American Mathematical Society

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