多变量全纯映射的动力学及其应用

This project studies the dynamics of holomorphic maps in several complex varaibles and its applications. In particular, we study the following questions: 1. The local invariants and attracting dynamics of holomorphic maps in several complex varaibles; 2. The iteration theory of holomorphic self-maps of bounded symmetric domains; 3. The applications of methods and ideas of dynamical systems in several complex variables. This project mainly studies complex dynamics in several variables, and its close connections with several complex variables and complex geometry. Complex dynamics in several variables is a relatively young research field and has seen rapid development in past decades. The content of this project covers several leading and new questions in the field, with few known results and lack of systematic studies. The research output of this project can hopefully boast the development of related theories of complex dynamics in several variables, and thus is of great academic value.

本项目研究多变量全纯映射的动力学及其应用。具体研究以下几方面的课题：1、多变量全纯映射的局部不变量及其吸性动力学；2、有界对称域上全纯自映射的迭代理论；3、动力系统方法与思想在多复变函数论中的应用。本项目的主体研究内容为多变量复动力系统，同时与多复变函数论以及复几何紧密结合。多变量复动力系统作为一个相对年轻的研究领域，在过去几十年发展迅速。本项目研究内容为该领域较前沿和新颖的课题，已知结果较少，且缺乏系统的研究方法。本项目的研究成果有望有效地推动多变量复动力系统相关理论的发展，有重要的学术价值。

本项目主要研究高维全纯映射的动力学及其在有界域上的函数论中的应用。在局部动力学方面，得到TTI映射的Fuchsian特征方向的一些新成果；在有界域上的动力学方面，对单位球上Wolff点的局部动力学进行了较完整的研究，同时系统研究了广义squeezing函数与Fridman不变量；在动力学应用方面，通过引进一维复动力系统的一些技巧得到Burns-Krantz型边界刚性定理的一些推广。这些成果均推动了相关课题的最新研究发展，具有较高的学术价值。