Study of functional equations by applying value distribution theory and complex dynamics

应用值分布理论和复杂动力学研究函数方程

• 批准号: 14540166
• 负责人: TOHGE Kazuya
• 金额: $ 2.62万
• 依托单位: KANAZAWA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540166/
• 关键词: Fermat type functional equation; structurally finite entire function; Painleve transcendents; complex dynamics; complex error function; q-difference equation; value distribution theory; wandering domain; quasiconformal surgery; Painlev\'e超越函数; 超

This research project has been carried out as planned, and each researcher who took responsibility for part of this project has achieved satisfactory results as follows :(0)Tohge studied some complex differential or functional equations and their solutions given as meromorphic functions in the plane. A new knowledge for the existence of those solutions was obtained and known estimates on the value distribution of those functions were sharpened considerably. He also considered how the facts obtained in this research relate to other subjects and found a guide for further researches.(1)Taniguchi studied the complex dynamics of entire functions, especially those of structurally finite entire functions in detail and succeeded to geometrize them combinatorically. He also investigated the structures of the spaces of Bell representations and the modified spaces of structurally finite entire functions, and studied related covering structure and dynamical structure of those functions.(2)Shimomur … More a studied function theoretic properties of the Painleve transcendents and gave the sharp estimates for the growth orders in both directions. He also observed the value distribution of fourth Painleve PIV concerning its moving targets.(3)Morosawa generalized the concept of semi-hyperbolic for rational functions into transcendental entire functions and also introduced the "complex" error function in order to investigate its dynamical properties at large and obtain new information on the figure of Julia sets, non-existence of wandering domains and Baker domains of those functions. He also obtained a result concerning the convergence of the sequence of Fatou sets of polynomials converging uniformly to a transcendental entire function.(4)Ishizaki studied various types of functional equations, which possess meromorphic solutions in the whole complex plane. Especially, he proved the existence of those solutions to linear difference equations and q-difference equations, as well as an exact estimate for their growth order. Then a new method, which seems to be applicable widely, was developed in the process.(5)Kisaka concentrated on the study of complex dynamics of entire functions, and succeeded to construct an example of transcendental functions, which possess a doubly-connected wandering domain by the method of quasi-conformal surgery. He also gave an example of those functions with a wandering domain of arbitrary assigned connectivity and settled this existence problem posed by I.N.Baker about 40 years ago Less

This research project has been carried out as planned, and each researcher who took responsibility for part of this project has achieved satisfactory results as follows :(0)Tohge studied some complex differential or functional equations and their solutions given as meromorphic functions in the plane. A new knowledge for the existence of those solutions was obtained and known estimates on the value distribution of those functions were sharpened considerably. He also considered how the facts obtained in this research relate to other subjects and found a guide for further researches.(1)Taniguchi studied the complex dynamics of entire functions, especially those of structurally finite entire functions in detail and succeeded to geometrize them combinatorically. He also investigated the structures of the spaces of Bell representations and the modified spaces of structurally finite entire functions, and studied related covering structure and dynamical structure of those functions.(2)Shimomur … More a studied function theoretic properties of the Painleve transcendents and gave the sharp estimates for the growth orders in both directions. He also observed the value distribution of fourth Painleve PIV concerning its moving targets.(3)Morosawa generalized the concept of semi-hyperbolic for rational functions into transcendental entire functions and also introduced the "complex" error function in order to investigate its dynamical properties at large and obtain new information on the figure of Julia sets, non-existence of wandering domains and Baker domains of those functions. He also obtained a result concerning the convergence of the sequence of Fatou sets of polynomials converging uniformly to a transcendental entire function.(4)Ishizaki studied various types of functional equations, which possess meromorphic solutions in the whole complex plane. Especially, he proved the existence of those solutions to linear difference equations and q-difference equations, as well as an exact estimate for their growth order. Then a new method, which seems to be applicable widely, was developed in the process.(5)Kisaka concentrated on the study of complex dynamics of entire functions, and succeeded to construct an example of transcendental functions, which possess a doubly-connected wandering domain by the method of quasi-conformal surgery. He also gave an example of those functions with a wandering domain of arbitrary assigned connectivity and settled this existence problem posed by I.N.Baker about 40 years ago Less

项目成果

K.Ishizaki, N.Yanagihara: "Deficiency for meromorphic solutions of Schroder equations"Complex Variables. (印刷中). (2004)

K.Ishizaki、N.Yanagihara：“施罗德方程亚纯解的缺陷”复变量（出版中）。

S.Snimomura: "On the number of poles of the first Painlev\'e transcendents and higher order analogues II"数理解析研究所講究録. 1316巻. 13-18 (2003)

S.Snimomura：“关于第一 Painleve 超越数和高阶类似物的极数 II”数学科学研究所 Kokyuroku Vol. 13-18 (2003)

K.Ishizaki: "A note on the functional equation $f^n+g^n+h^n=1$ and some complex differential equations"Computational Method and Functional Theory. (to appear).

K.Ishizaki：“关于函数方程 $f^n g^n h^n=1$ 和一些复杂微分方程的注释”计算方法和泛函理论。

W.Bergweiler, S.Morosawa: "Semihyperbolic entire functions"Nonlinearity. 15. 1673-1684 (2002)

W.Bergweiler、S.Morosawa：“半双曲整函数”非线性。

K.Ishizaki: "A note on Factorization of the Weierstrass $Pe$-function"Proceedings of the 3rd ISAAC Congress. (発表予定).

K. Ishizaki：“关于 Weierstrass $Pe$ 函数分解的说明”第三届 ISAAC 大会记录（待提交）。