Analytic transformations of complex manifolds

作为单复变量有理函数迭代理论的推广，我们研究了由复射影空间在自身上的全态映射所定义的复动力系统。Fatou集被定义为这样一个全纯映射的迭代族构成正规族的最大开集。这被认为是该理论最基本的对象之一。在本文的研究中，我们证明了Fatou集是伪凸集，因此是一个Stain开集，并进一步证明了每个吸引周期点或抛物周期点的吸引盆都包含一个临界点。我们也给出了在射影平面上的动力系统的一些例子，其中Fatou集可以用椭圆函数具体描述，并且Fatou集是空的。进一步研究了Fatou集、临界点集的正轨道及其极限集集之间的关系。特别地，我们研究了临界有限情况，即临界点集合的轨道是代数集合的情况。对于维度2的情况，我们已经给出了Fatou集合为空的条件。

We investigeted complex dynamical system defind by holo-morphic maps of a complex projective space onto itself, as a generalization of the iteration theory of rational function of one complex variable.The Fatou set is defined to be the maximal open set on which the family of the iterates of such a holomorphic map constitute a normal family. This is considered as one of the most fundamental object in the theory. In our study we have proved that the Fatou set os pseudoconvex and hence a Stain open set, and further that follows that, every basin of attraction of an attracting periodic point or that of parabolic periodic point contains a critical point.We have also given some examples of dynamical systems ori pro-jective planes for which the Fatou set can be concretely described using elliptic functions, and for which the Fatou set is empty.Further we studied the ralation among the Fatou set, the forward orbit of the set of the critical points and its limit set set. In particular, we stydied the critically finite case, i.e., the case for which the orbit of the the set of the critical points is an algebraic set. For the case of dimension 2, we have given the condi-tion for the Fatou set is empty.