ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

doi:10.1007/s10473-020-0401-5

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (4): 903-909.doi: 10.1007/s10473-020-0401-5

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ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

曹春雷¹, 王跃飞^2,3

1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
2. College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;
3. Academy of Mathemtics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China

收稿日期:2019-07-08 出版日期:2020-08-25 发布日期:2020-08-21
通讯作者: Yuefei WANG E-mail:wangyf@math.ac.cn
作者简介:Chunlei CAO,E-mail:clcao@bit.edu.cn
基金资助:
The work was supported by NSF of China (11688101).

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

Chunlei CAO¹, Yuefei WANG^2,3

1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
2. College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;
3. Academy of Mathemtics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China

Received:2019-07-08 Online:2020-08-25 Published:2020-08-21
Contact: Yuefei WANG E-mail:wangyf@math.ac.cn
Supported by:
The work was supported by NSF of China (11688101).

摘要/Abstract

摘要： In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of ${\bf C^*}$ can neither contain a free Jordan arc nor be contained in any finite set of straight lines.

关键词: Fatou set, Julia set, transcendental analytic map, free Jordan arc

Abstract: In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of ${\bf C^*}$ can neither contain a free Jordan arc nor be contained in any finite set of straight lines.

Key words: Fatou set, Julia set, transcendental analytic map, free Jordan arc

中图分类号:

37F10

曹春雷, 王跃飞. ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS[J]. 数学物理学报(英文版), 2020, 40(4): 903-909.

Chunlei CAO, Yuefei WANG. ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS[J]. Acta mathematica scientia,Series B, 2020, 40(4): 903-909.

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ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

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