JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

doi:10.1007/s10473-021-0415-7

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (4): 1275-1286.doi: 10.1007/s10473-021-0415-7

JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

王珺¹, 姚潇², 张诚纯²

1. School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

收稿日期:2020-05-04 出版日期:2021-08-25 发布日期:2021-09-01
通讯作者: Xiao YAO E-mail:yaoxiao@nankai.edu.cn
作者简介:Jun WANG,E-mail:majwang@fudan.edu.cn;Xiao YAO,E-mail:yaoxiao@nankai.edu.cn;Chengchun ZHANG,E-mail:18210180014@fudan.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (11771090, 11901311) and Natural Sciences Foundation of Shanghai (17ZR1402900).

JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

Jun WANG¹, Xiao YAO², Chengchun ZHANG²

1. School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received:2020-05-04 Online:2021-08-25 Published:2021-09-01
Contact: Xiao YAO E-mail:yaoxiao@nankai.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (11771090, 11901311) and Natural Sciences Foundation of Shanghai (17ZR1402900).

摘要/Abstract

摘要： For entire or meromorphic function $f$, a value $\theta\in[0,2\pi)$ is called a Julia limiting direction if there is an unbounded sequence $\{z_n\}$ in the Julia set satisfying $\lim\limits_{n\rightarrow\infty}\arg z_n=\theta.$ Our main result is on the entire solution $f$ of $P(z,f)+F(z)f^s=0$, where $P(z,f)$ is a differential polynomial of $f$ with entire coefficients of growth smaller than that of the entire transcendental $F$, with the integer $s$ being no more than the minimum degree of all differential monomials in $P(z,f)$. We observe that Julia limiting directions of $f$ partly come from the directions in which $F$ grows quickly.

关键词: Julia set, meromorphic function, Julia limiting direction, complex differential equations

Abstract: For entire or meromorphic function $f$, a value $\theta\in[0,2\pi)$ is called a Julia limiting direction if there is an unbounded sequence $\{z_n\}$ in the Julia set satisfying $\lim\limits_{n\rightarrow\infty}\arg z_n=\theta.$ Our main result is on the entire solution $f$ of $P(z,f)+F(z)f^s=0$, where $P(z,f)$ is a differential polynomial of $f$ with entire coefficients of growth smaller than that of the entire transcendental $F$, with the integer $s$ being no more than the minimum degree of all differential monomials in $P(z,f)$. We observe that Julia limiting directions of $f$ partly come from the directions in which $F$ grows quickly.

Key words: Julia set, meromorphic function, Julia limiting direction, complex differential equations

中图分类号:

34M05

王珺, 姚潇, 张诚纯. JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(4): 1275-1286.

Jun WANG, Xiao YAO, Chengchun ZHANG. JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia,Series B, 2021, 41(4): 1275-1286.

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JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

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