复微分方程和复动力系统及其应用的研究是当代数学研究的热点之一，吸引了很多著名的国内外复分析学者的关注。本项目将利用复动力系统理论、拟共形映射理论、渐近分析理论、Nevalinna理论、Wiman-Valiron理论、正规族理论、函数唯一性理论中的思想方法研究复域上微分方程亚纯解的解析性质，如代数微分方程解的增长级、Painleve方程及其高阶类似的亚纯解的值分布性质、四类Painlevé差分方程解的零点分布和增长性；另一方面研究Painleve方程、线性微分方程和线性差分方程解的动力学性质，探讨其亚纯解的无界Fatou分支、Julia集径向分布和Hausdorff维数及其不变测度。

Value distribution theory, complex dynamics and their intersection are the important topics in the modern mathematics, which have also attracted the attention of many famous mathematicians in complex analysis. Our project aims to discuss analytic properties and dynamics behavior of meromorphic solutions of complex differential equations and complex difference equations, and also study the analytic, geometric and topological property of the Julia sets of rational functions. We would apply the theories in complex dynamics, quasi-conformal mapping, asymptotic analysis and value distribution. By using the development and the method in complex dynamics, we try to investigate some problems in Painlevé equations, difference equations, value distribution and statistical physics.