Deformation theory of Kleinian groups

克莱因群的变形理论

• 批准号: 09640162
• 负责人: WATANABE Hisako
• 金额: $ 1.47万
• 依托单位: Ochanomizu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640162/
• 关键词: Kleinian group; Fuchsian group; Hausdorff dimension; Teichmuller space; quasiconformal map; deformation; fractal set; dynamical system; 複素力学系; 2重層ポランシャル; ホロノミー写像; wandering domain; フラクタル; 2重層ポテンシャル

1. By geometric properties of corresponding a hyperbolic manifold we described the necessary conditions in order that the Hausdorff dimension of the limite of a n-dimensinal hyperbolic discrete group is less than n.2. We investigated the continuity of the Hausdorff dimensions as the Kleinian groups are deformed.3. We analized the structures of the set of discrete representations in the parameter space and proposed the method of analizing the structure of the space of quasi-Fuchsian groups by a holonomy map from the space of projections on a Riemann manifold.4. We obtained the results on the structural stability of Kleinian groups under small perturbations and on the equivalence between the algebraic topology and the Teichm_ller one in the space of quasiconformal deformations.5. We built the deformation theory of dynamical systems for entire functions on a ground of Teichm_ller space and found the fundamental properties of wan-dering domains and Baker domains, which rational functions don't have.6. We investigated a family of inlaid functions and showed the topological completeness of it. Further we found the Teichm_ller spaces of the Fatou components of those functions.

1.利用双曲流形的几何性质，给出了n维双曲离散群的极限的Hausdorff维数小于n.2的必要条件.我们研究了当Kleinian群变形时Hausdorff维数的连续性.分析了参数空间中离散表示集的结构，提出了从黎曼流形上的投影空间出发，利用完整映射分析拟Fuchsian群空间结构的方法.得到了Kleinian群在小扰动下的结构稳定性以及拟共形变形空间中代数拓扑与Teichm_ller拓扑的等价性.在Teichm_ller空间的基础上建立了整函数动力系统的形变理论，发现了有理函数不具有的游荡域和Baker域的基本性质.本文研究了一类镶嵌函数族，证明了它的拓扑完备性，并给出了这类函数的Fatou分支的Teichm_ller空间。

项目成果

M.Taniguchi: "Logarithmic lifts of the family λze^Z" 京都大学数理解析研究所講究録. 988. 21-28 (1997)

M.Taniguchi：“族 λze^Z 的对数升力”京都大学数学科学研究所 Kokyuroku。988. 21-28 (1997)

M.Taniguchi: "Logarithmic lifts of the family lambdaze^Z" RIMS koukyuroku. 988. 21-28 (1997)

M.Taniguchi：“家族 lambdaze^Z 的对数升力”RIMS koukyuroku。

K.Matsuzaki: "Hausdorff dimension of the limit sets of infinitely generated Kleinian groups" Math.Proc.Cambridge Phil.Soi.128 (To appear).

K.Matsuzaki：“无限生成克莱因群的极限集的豪斯多夫维数”Math.Proc.Cambridge Phil.Soi.128（待出现）。

H.Watanabe: "The initial-boundary value problems for the heat operator in non-cylindrical domains" J.Math.Soc.Japan. 49・3. 399-430 (1997)

H. Watanabe：“非圆柱域中热算子的初始边界值问题”J.Math.Soc.Japan 49・3（1997）。

H.Watanabe: "Double layer potentials of functions in a Besov space for a bounded domain with fractal boundary" Proceedings of the Fifth International Conference on Complex Analysis. (in press).

H.Watanabe：“具有分形边界的有界域的贝索夫空间中函数的双层势”第五届国际复分析会议论文集。