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Analysis and Applications of Teichmuller space

Teichmuller空间的分析与应用

基本信息

批准号：
08454026
负责人：
TANIGUCHI Masahiko
金额：
$ 4.54万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至无数据
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454026/
关键词：
Teichmuller space Complex dynamics Kleinian groups Quasiconformal map Riemann surface Fractal sets Julia sets Hausdorff dimension

项目摘要

The results can be devided into two categories ; those on the Teichmuller spaces and those on objects which provide the Teichmuller spaces. Among oter things, the research on the Teichmuller spaces of infinite dimension is very important.The Head investigator Taniguchi has clarified the fundamental structures of the Teichmuller space of a transcendental entire function, including the relationship between the absence of wandering domains and finite dimensionality of the corresponding Teichmuller space. These results are the coproduct with T.Harada, a graduate student, and very important, for they give a new light for the investigations on the complex dynamics induced by transcendental entire functions as in the cases of rational functions and of Kleinian groups.Also Taniguchi has investigated the coiling property, appeared only in the case of infinite dimension, and proved that the Bloch convergence on the universal Teichmuller space is equivalent to Caratheodory convergence, when we consider points of the universal teichmuller space as fractal sets on the plane.Next, a crucial divice for the Teichmuller theory is a quasiconformal map. Investigator Sugawa has suceeded to give a characterization of the Teichmuller spaces of finite Riemann surfaces without cusps. Sugawa has also given quantative estimate on domain constants related to uniform perfectness. As a consequence, we have interesting estimates of the Hausdorff dimensions of the Julia sets of a complex dynamics, or of the limit set of a Kleinian group.

结果可分为两类；那些在泰希米勒空间上的以及那些在提供泰希米勒空间的物体上的。其中，对无限维Teichmuller空间的研究非常重要。课题组长谷口阐明了超越整函数的Teichmuller空间的基本结构，包括不存在游走域与相应Teichmuller空间的有限维数之间的关系。这些结果是与研究生 T.Harada 的联产品，非常重要，因为它们为研究由超越整体函数引起的复杂动力学（如有理函数和 Kleinian 群的情况）提供了新的思路。此外，Taniguchi 还研究了仅在无限维情况下出现的卷曲性质，并证明了普适 Teichmuller 空间上的 Bloch 收敛性等价于当我们将普适泰希米勒空间的点视为平面上的分形集时，卡拉西奥多里收敛。接下来，泰希米勒理论的一个关键设备是拟共形映射。研究员菅川成功地描述了无尖点的有限黎曼曲面的 Teichmuller 空间的特征。菅川还对与均匀完美性相关的域常数给出了定量估计。因此，我们对复杂动力学的 Julia 集或 Kleinian 群的极限集的豪斯多夫维数进行了有趣的估计。