代数体函数的幅角分布与迭代

Influenced by the Ahlfors’ theory of covering surfaces and the potential theory, the angular distribution and the iteration of algebroid functions have become a central issue that draws many researchers'' attention. Algebroid functions are the extention of meromorphic functions, the theory of these functions helps to enrich the development of other subjects such as complex analysis and complex differential equations. . In this project, we will study the following aspects: Firstly, we will investigate the angular distribution of algebroid functions. By establishing the Nevanlinna’s Characteristic or Tsuji’s Characteristic in an angle, we can describe the releationship between the angular distribution and the order. Secondly, we will discuss the domains containing some singular directions, the uniqueness dealing with deficient values. Lastly, we will study the dynamics of the random iteration of transcendental entire functions. In view of this and the classification theorem of algebroid functions, we can define and investigate the escaping point(set) of entire algebroid functions.. In previous researches, we have proved the existence of singular directions, some uniqueness theorems dealing with deficient values and have initially described the topological structure of the escaping points of entire functions. In this project, we will make an intensive study of the angular distribution and the iteration of algebroid functions.

近年来，随着Ahlfors覆盖曲面理论及位势论的渗透，代数体函数的幅角分布和迭代已成为一个备受关注的研究热点。代数体函数是亚纯函数的推广，为复分析和微分方程理论的发展注入了活力。本项目将研究代数体函数的幅角分布，尝试建立角域内代数体函数的Nevanlinna特征函数或是Tsuji特征函数，进而研究幅角分布和增长级之间的关系；对奇异方向的存在区域进行刻画，在存在亏值条件下研究角域内的唯一性问题；发展有限个整函数的随机迭代动力学，结合整代数体函数的分类定理定义并研究整代数体函数的逃逸点(集)。在之前的研究中，我们已经证明了奇异方向的存在性并对全平面上涉及亏量的代数体函数唯一性进行了刻画；对整函数的逃逸集的层结构进行了初步刻画。本项目将继续深化代数体函数的幅角分布和迭代的研究。

本项目研究了角域内代数体函数的特征函数、唯一性问题，整代数体函数的逃逸集问题。对代数体函数结式零点进行仔细估计，对其在角域内的数量恰当界定，从而可以控制分支点的计数函数。尝试建立代数体函数角域上的Nevanlinna特征函数、Tsuji特征函数。利用Polya峰和单位圆内代数体函数的第二基本定理研究代数体函数在角域内涉及亏值的唯一性问题，定义整代数体函数的逃逸集并建立逃逸集的基本性质。另外，我们还研究了1-值代数体函数（亚纯函数）的奇异方向问题及右半平面上Laplace-Stieltjes定义的无穷级解析函数的值分布问题。