Mathematical Proceedings of the Cambridge Philosophical Society



On uniformly perfect boundary of stable domains in iteration of meromorphic functions II


ZHENG JIAN-HUA a1 1
a1 Department of Mathematical Sciences Tsinghua University, Beijing, P.R. China. e-mail: jzheng@math.tsinghua.edu.cn

Abstract

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

(Received August 10 2000)
(Revised January 8 2001)



Footnotes

1 This work was supported by NSF of China.