Dynamics of modular groups on infinite dimensional Teichmuller spaces

Teichmueller空间不是齐性空间，它们的模群不传递。对于紧致黎曼曲面，模群的作用是不连续的，但对于无限维的Teichmueller空间，情况并非如此。通过考虑模群作用的混沌行为，研究了无限型黎曼曲面的模空间。从一般拓扑学的观点看，模空间要么是第一分离公理可度量化的，要么不是。然而，除了单个部分之外，它可以具有一定的几何结构。在本研究中，我们利用黎曼曲面的双曲几何结构来刻画这个稳定区域，并利用这个稳定区域的完备化来构造一个压缩模空间。因此，我们可以用黎曼曲面的几何特征来描述点集的闭包，本文考虑了单位圆盘上单叶函数的预Schwarzian导数空间，它扩张到扩张平面上的拟共形映射，以研究预Schwarzian导数空间的连通分支之间的关系。单位圆盘上单叶函数的Schwarzian导数，推广到扩张平面上的拟共形映射，以研究泛Teichmueller空间的Pre-Schwarzian模型的连通分支与单叶函数的经典族之间的关系.我们还研究了具有Schwarzian导数的指定增长的单叶函数的几何性质，并发现它们是星形的或凸的，根据到原点的距离，在通用Teichmueller空间的Bers嵌入。

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.