复动力系统及复分析中若干问题的研究

复动力系统是现代数学研究的重要分支之一。复动力系统的研究涉及多个数学物理分支。本项目主要研究复动力系统中不稳定集的拓扑和几何性质以及复动力系统在复分析和物理中的相关问题。具体问题如下：通过对淹没点集的研究进一步刻画Julia集的复杂性，建立新的统计物理学模型，利用复动力系统方法研究Yang-Lee问题与玻璃态相变；通过隐藏周期轨道研究高维复动力系统中中性周期点的局部几何性质及其应用；涉及复微分方程中的动力系统性质的研究及Teichmuller流。本研究课题属于复动力系统及复分析中的核心问题，具有重要的理论意义。

Complex dynamics is an important field of morden mathematics. And it is related with several fields of mathematics and physics. This project focuses the topology and geometry of unstable sets and some problems arised in complex analysis and physics，which are related with complex dynamics. The problems are as follows: give characteristics of Julia sets by the set of buried points; give some new statistical physics models and investigate the problems about the zeros of Yang-Lee and glass phase transition by complex dynamics; give the local geometry of indifferent periodic points by hidden periodic orbits in higher dimensional complex dynamics and its applications; give the dynamical properties of complex differential equations and Teichmuller fow. This project is on the central problems of complex dynamics and it plays an important role in the theory of complex dynamics and related fields.