Sullivan 定理在一类广义亚纯函数中的推广

In this program we will investigate the dynamics of a class of generalized meromorphic functions which are meromorphic outside a compact totally disconnected set containing infinitely many points. We will mainly study the Fatou set of functions in the above class .and prove the following two propositions: 1. any function in the above class with finitely many singular values has no wandering Fatou component. 2. any function in the above class satisfying that the set of its singular values is bounded and the Euclidean distance between the set of its singular values and its Julia set is positive has no wandering Fatou component.

本项目研究一类更广泛的亚纯函数动力系统，也就是在一个完全不连通并且含有无限多个点的紧集之外亚纯的函数。我们主要研究上述函数的Fatou集，拟证明如下两个命题：1. 只有有限个奇异值的上述函数Fatou集没有游荡分支。2.奇异值集有界且其前向轨道与Julia 集的欧式距离为正的上述函数Fatou集没有游荡分支。

本项目组研究了一族M类亚纯函数的动力系统。首先，研究了该亚纯函数族的M-集在实轴上的拓扑结构，证明了它的M-集在实轴上的补集是一个有限的左开右闭区间；.其次，研究了实参数的该族亚纯函数的Fatou集的拓扑性质，证明了当参数属于上述有限区间（除去右端点）时，对应亚纯函数的Fatou集由无穷多个单连通分支构成，并且没有完全不变的连通分支；最后，研究了实参数的该族亚纯函数的Julia集的拓扑性质，证明了当该族M类亚纯函数含有吸性周期循环时，则其Julia集中几乎所有点的前向轨道的极限点都是原点和无穷远。