Dynamics of modular groups on infinite dimensional Teichmuller spaces

Teichmueller空间不是齐次空间，其Mudular群不是传递性的。对于紧黎曼曲面，模群的作用是不连续的，但对于无限维Teichmueller空间则不是这样。通过考虑模群作用的混沌行为，研究了无限型Riemann曲面的模空间。从一般拓扑学的观点来看，模空间要么是第一分离公理的可度量化的，要么是不可度量化的。然而，除了奇异的部分，它可以具有一定的几何结构。本文利用Riemann曲面的双曲几何结构刻画了这一稳定区域，并利用该稳定区域的完备性构造了一个压缩模空间。考虑了单位圆盘上单叶函数的拟共形映射空间，研究了单位圆盘上单叶函数的拟共形映射的连通分支与扩展平面的拟共形映射的关系，研究了普适Teichmueller空间的单叶函数准Schwarzian模型的连通分支与经典单叶函数族之间的关系.我们还研究了具有给定Schwarzian导数增长量的单叶函数的几何性质，发现它们根据到原点的距离在泛Teichmueller空间的Bers嵌入中是星形的或凸的。

Teichmueller spaces are not homogeneous spaces and their mudular groups do not act transitively. For compact Riemann surfaces, modular groups act discontinuously, but this is not the case for infinite dimensional Teichmueller spaces. We study the moduli spaces of Riemann surafces of infinite type by considering the chaotic behavior of the action of modular groups. For a viewpoint of general topology, the moduli space is either metrizable or not of the first separation axiom. However, except for a singular part, it can possess a certain geometric structure. In this research, we characterize this stable region by hyperbolic geometric structure of a Riemann surface and construct a contracted moduli space by the completion of the stable region. Consequently, we can describe the closure of a point set in terms of the geomery of Riemann surfaces, which is a point of teh contracted module space.We considered the space of pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian model of the universal Teichmueller space and classical families of univalent functions. We also investigated geometric properties of univalent functions with a prescribed growth of the Schwarzian derivative and found that they are starlike or convex according to the distance to the origin in the Bers embedding of the universal Teichmueller space.