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ON THE ITERATIONS AND THE ARGUMENT DISTRIBUTION OF MEROMORPHIC FUNCTIONS

Part of: Entire and meromorphic functions, and related topics Complex dynamical systems

Published online by Cambridge University Press:  15 February 2024

JIE DING  and
JIANHUA ZHENG
JIE DING*
Affiliation:
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, PR China
JIANHUA ZHENG
Affiliation:
Department of Mathematical Science, Tsinghua University, Beijing 100084, PR China e-mail: zheng-jh@mail.tsinghua.edu.cn
*
e-mail: dingjie@tyut.edu.cn
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Abstract

This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

Keywords

meromorphic functionFatou setJulia setBorel directionfilling disk

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 116 , Issue 2 , April 2024 , pp. 145 - 170
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Milena Radnovic

The first author was supported by Grant No. 202103021224069 of Shanxi Basic Research Program.

The second author was supported by Grant No. 12071239 of the NSF of China.

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