本项目研究一个亚纯函数族的动力系统，该函数族恰好有一个Picard 例外值的极点0，主要研究两个课题。第一个课题是关于上述亚纯函数族(0是一阶极点)的类Mandelbrot集，这里Mandelbrot集是所有使得对应映射的临界点前向轨道有界的参数构成的集合。在该课题中，我们拟证明如下两个命题：1. 上述类Mandelbrot集的每个双曲分支U在复平面上的边界是Jordan曲线，这里双曲分支是双曲参数集的连通分支，双曲参数是使得对应函数含有吸性周期循环的参数。命题2. 上述类Mandelbrot集在每个双曲分支U位于复平面上的边界点和Misiurewicz参数处都是局部连通的，这里Misiurewicz参数是使得 的奇异值是预周期点的参数。第二个课题是关于实参数的上述亚纯函数族(0是任意阶极点)的刚性问题。在该课题中，我们拟建立实参数的上述亚纯函数族关于组合等价性的刚性定理。

In this program we will study the dynamics of a family of meromorphic functions with exact one pole, which is a Picard exceptional value. It includes two topics. The first topic is concerning the Mandelbrot-like set of the above family with the pole 0 of first-order, where the Mandelbrot-like set consists of all parameters whose corresponding map has bounded critical forward orbit. In this topic, we shall prove the following two propositions: 1. The boundary of every hyperbolic component of the Mandelbrot-like set on the complex plane is a Jordan curve, where a hyperbolic component is a connected component of the set of all hyperbolic parameters whose corresponding maps have attracting cycles. 2. The Mandelbrot-like set is locally connected at each point on the boundary of each hyperbolic component and each Misiurewicz point, where Misiurewicz points are the parameters whose corresponding maps have preperiodic singular values. The second topic is concerning the rigidity of the above family with the pole 0 of arbitrary order and real parameters. In this topic, we plan to establish a rigidity theorem about combinatorial equivalence of the above family with real parameters.