本项目致力于研究有理函数动力系统领域的重要问题，包括以下内容。 1. 刻画有理函数非旋转Fatou域的共形结构，包括对抛物域构造全纯模型；探讨非旋转域全纯模型的拓扑性质及可测动力系统；考察非旋转域游荡边界分支的拓扑性质。 2. 对Julia集是Cantor集的有理函数及不可重整多项式讨论抛物点扰动问题。 3. 研究有理函数不连通Julia集的组合结构和拓扑性质，包括对给定度大于2，尝试构造次双曲有理函数，使得它的复杂型Julia分支循环的个数是任意正整数；对具有不连通Julia集的一般有理函数，研究始终是复杂型游荡分支的存在性问题等。 4. 研究Sierpinski有理函数双曲分支的预紧性。尝试利用folding手术构造双曲分支预紧的临界有限Sierpinski有理函数；对一般的Sierpinski有理函数，研究其双曲分支的分析性质，期望解决Sierpinski有理函数双曲分支预紧性问题。

In this project, we will investigate some important problems in the field of the dynamics of rational maps. The project focuses on the following items. 1. Study the conformal structure of parabolic Fatou domains and try to construct a holomorphic model for a parabolic domain. Study the measurable dynamical systems and the topological properties of the holomorphic models of non-rotation Fatou domains. 2. Discuss the perturbation about the parabolic points of the rational maps with Cantor Julia sets and non-renormalizable polynomials. 3. Investigate the combinatorics and the topology of the complex type Julia components. Hope to prove that given degree bigger than two and a positive integer, there exsits a sub-hyperbolic rational map with the given degree such that the number of the cycles of complex type Julia components is the given integer. 4. Study the precompactness of Sierpinski hyperbolic components. Try to construct a Sierpinski rational with a precompact hyperbolic component by means of folding surgery. Discuss the analytic properties of Sierpinski hyperbolic components.