高维解析映射的局部吸性动力学及整体动力结构

This project studies the local attracting dynamics and global dynamical structures of holomorphic maps in higher dimensions. In particular, we study the following questions: 1. The attracting dynamics in degenerate characteristic directions of tangent to the identity maps in dimension two; 2. Local invariants and attracting dynamics of tangent to the identity maps in higher dimensions; 3. The Fatou set of holomorphic maps on complex projective spaces in higher dimensions; 4. The period structure of holomorphic maps on complex manifolds in higher dimensions. This project belongs to complex dynamics in several variables. Complex dynamics in several variables is a relatively young brach of several complex variables 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 both the local and global theories of complex dynamics in several variables, and thus is of great academic value.

本项目研究高维解析映射的局部吸性动力学及整体动力结构。具体研究以下几方面的课题：1、二维tangent to the identity映射在退化特征方向的吸性动力学；2、高维tangent to the identity映射的局部不变量与吸性动力学；3、高维复投影空间上解析映射的Fatou集结构；4、高维复流形上解析映射的周期结构。本项目研究属于多变量复动力系统范畴。多变量复动力系统是多复变函数论领域一个相对年轻的分支，在过去几十年发展迅速。本项目研究内容均为该领域非常前沿和新颖的课题，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局部理论与整体理论的发展，有重要的学术价值。

本项目主要研究了全纯映射的动力学性质，并延伸到研究了有界域上的分析与几何。在本项目执行过程中，我们取得较可观的研究成果，特别是提出了数个新的概念，为今后相关课题的研究指明了方向。比较重要的成果包括：1.提出全纯映射局部不变量的“层次”概念；2.定义了诸如non-dicritical order, essential order等新的局部不变量；3.提出resonance order的概念，进而推广了Cartan线性化定理；4.成功将动力学结果应用于有界域的研究。