Dynamics of modular groups on infinite dimensional Teichmuller spaces

Teichmueller空间不是齐次空间，其模群不具有传递性。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空间却不是这样。考虑模群作用的混沌行为，研究无穷型黎曼曲面的模空间。对于一般拓扑的观点，模空间要么是可度量的，要么是第一分离公理不可度量的。但是，除了单个部分外，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表征该稳定区域，并通过该稳定区域的补全构造一个收缩模空间。因此，我们可以用黎曼曲面的几何来描述点集的闭包，它是压缩模空间中的一个点。我们考虑了单位盘上扩展到扩展平面的拟共形映射的一元函数的预schwarzian导数的空间，以研究单位盘上扩展到扩展平面的拟共形映射的一元函数的预schwarzian导数的连通分量之间的关系，从而研究了通用Teichmueller空间的预schwarzian模型的连通分量与一元函数的经典族。我们还研究了具有Schwarzian导数规定增长的一元函数的几何性质，并发现它们在通用Teichmueller空间的Bers嵌入中根据到原点的距离是星形或凸形的。

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.