复解析动力系统是现代数学研究的主流方向之一，国际上有三位数学家因在复解析动力系统的研究获得Fields奖章。复动力系统有大量具有挑战性的获奖问题，其中心问题——双曲性猜想仍未解决。本申请项目将致力于将复动力系统关于拓扑动力系统的研究推进一步，对复解析动力系统的主要研究对象Julia集、Fatou分支、参数空间中的双曲分支、以及整函数的逃逸集的几何性质做进一步的刻画，内容涉及其拟共形（拟对称）几何和Hausdorff维数和测度的研究，主要研究内容包括Sierpinski地毯Julia集的拟对称单值化和拟对称刚性，捕获双曲分支的拟共形几何，Julia集的Hausdorff维数的渐近公式，以及某些整函数族的逃逸集和参数逃逸集的Hausdorff维数和gauge函数的刻画等内容。上述研究将有助于推动复解析动力系统的研究的深入发展。

Complex analytic dynamical systems is an important research field in morden mathematics. Three mathematicians won the Fields medal because of their researches in the field of complex analytic dynamical systems. There are a lot of challenged problems in this field, for example, the center problem - hyperbolic conjecture - remains open. This project will be devoted to pushing the research on topology of complex analytical dynamics into next step. We will give some geometric descriptions for the Julia sets, Fatou components, hyperbolic components in the parameter space, and the escape sets for entire functions, including their quasiconformal (quasisymmetrical) geometry and the Hausdorff dimension and measure. The main contents are the quasisymmetric uniformization and quasisymmetric rigidity of Sierpinski carpet Julia sets; the quasiconformal geometry of captured hyperbolic components, the asymptotic formula for the Hausdorff dimension of Julia sets, and the gauge function of Hausdorff measure of the escape sets and the parameter escape set for a family of entire functions. We hope that our work will contribute to the research on complex analytic dynamical systems.