Analysis of Complex Dynamics

首席研究员谷口先生为整个功能的覆盖结构引入了一种新的组合模型，称为“配置树”。这个概念对应于沙利文字典中Kleinian群的Cayley图。这个新模型使我们找到了一个非常重要的全函数类，它被称为结构有限全函数类。我们还发现，该类的每一个元素都有非常明确的表示为修饰指数函数的原始函数。因此，我们也证明了每一个结构有限完整函数的Julia集具有2维Hausdorff。在此基础上，利用全纯分支覆盖结构的一种表示空间——Hurwitz空间，肯定地证明了多连通平面域的显式表示的Bell猜想。

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.