复解析动力系统是现代数学研究的主流方向之一，有大量具有挑战性的问题有待解决，近年来分形集上的拟对称几何也取得了重要的进展。本项目将两者结合起来，研究有理函数动力系统中的Julia集和参数空间的拓扑和拟共形几何问题，主要研究：McMullen函数族参数平面的拓扑和非逃逸集的局部连通性，McMullen函数族及更一般有理函数的Julia集的拟共形分类以及拟对称单值化和刚性，Julia集的 Hausdorff维数和Ahlfors正则共形维数，一般的分形集合如Cantor拟圆周的拟对称单值化，以及某些度量空间的正则性研究。本项目将在上述研究中解决几个公开问题和猜想，如Devaney关于McMullen函数族参数平面的公开问题，并取在各个方面都取得一些突破性进展。

The complex analytical dynamics is one of the most important fields in morden mathematics. There are many challenging problems in this field to be solved. Recently, the quasisymmetric geometry on the fractal sets has been made important progress. This project will study problems on the topology and quasiconformal geometry of the Julia sets and parameter space of rational dynamics by combining these two fields. The researches include: the topology of parameter plane of McMullen rational maps; the quasiconformal classification of Julia sets of McMullen maps and their quasisymmetrical uniformization and rigidity; the Hausdorff dimensions and the Ahlfors regular conformal dimensions of Julia sets; the quasisymmetrical uniformization of fractal sets such as the Cantor quasicircles; and the regularities of some metric spaces. In this project, we will make some breakthroughs and solve some open problems, such as the problem posed by Devaney on the parameter plane of McMullen maps.