Analysis of Complex Dynamics

复杂动力学分析

• 批准号: 13440047
• 负责人: TANIGUCHI M
• 金额: $ 4.29万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440047/
• 关键词: Complex dynamics; Sullivan's dictionary; entire function; Kleinian group; representation space; branched cover

The head investigator, M. Taniguchi introduced a new kind of combinatorial model for the covering structure of an entire function, which is called "a configuration tree". This concept is the counterpart of that of the Cayley graph of a Kleinian group in the context of the Sullivan's dictionary.This new model enabled us to find a very important class of entire functions, which is called the class of structurally finite entire functions. We also discovered that every element of this class has the very explicit representation as a primitive function of a decorated exponential function. As a consequence, we also proved that the Julia set of every structurally finite entire function has the Hausdorff dimension two.Furthermore, very recently we proved the Bell conjecture on explicit representation of multi-connected planar domains affirmatively by using the Hurwitz space, which is a kind of the representation space of holomorphic branched covering structures.

带头研究人员M.Taniguchi为整个函数的覆盖结构引入了一种新的组合模型，称为“配置树”。这个概念对应于Sullivan词典中Kleinan群的Cayley图的概念，这个新的模型使我们能够找到一类非常重要的整函数，称为结构有限整函数类。我们还发现，这个类的每个元素都有非常明确的表示，作为修饰指数函数的本原函数。证明了结构有限整函数的Julia集的Hausdorff维为2，并利用全纯分支覆盖结构的表示空间--Hurwitz空间，肯定地证明了Bell猜想关于多连通平面域的显式表示.

项目成果

H.Shiga: "On holomorphic families of rational maps : finiteness, rigidity and stability"Kodai Math. J.. 24. 48-65 (2001)

H.Shiga：“论有理映射的全纯族：有限性、刚性和稳定性”Kodai Math。

Taniguchi, Masahiko: "Size of the Julia set of a structurally finite transcendental entire functio"Math. Proc. Camb. Phil. Soc.. to appear.

谷口正彦：“结构上有限的超越整个函数的朱莉娅集的大小”数学。

Taniguchi, Masahiko: "Explicit representation of structurally finite entire functions"Proc. Japan Acad. 77 Ser. A. 68-70 (2001)

Taniguchi，Masahiko：“结构上有限的整个函数的显式表示”Proc。

Gedeon, Tomavs, Kokubu, Hiroshi, Mischaikow, Konstantin, Hiroe, Oka: "Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing"J. Dynam. Differential Equations. 14. 63-84 (2002)

Gedeon、Tomavs、Kokubu、Hiroshi、Mischaikow、Konstantin、Hiroe、Oka：“哈密顿系统缓慢变化扰动中的混沌解及其在浅水晃动中的应用”J。

K.Matsuzaki: "Simply connected domains on a hyperbolic surface"New Zealand J. Math.. (to appear).

K.Matsuzaki：“双曲曲面上的单连通域”新西兰 J. Math..（即将出现）。