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Deformation theory of Kleinian groups

Research Project

Project/Area Number 09640162
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionOchanomizu University

Principal Investigator

WATANABE Hisako  Ochanomizu University, Faculty of Science, Professor, 理学部, 教授 (70017193)

Co-Investigator(Kenkyū-buntansha) MAEDA Michie  Ochanomizu University, Faculty of Science, Professor, 理学部, 教授 (30017206)
TANIZAKI Masahiko  Kyoto University, Research Course in Science, Assistant Professor, 大学院・理学研究科, 助教授 (50108974)
MATSUZAKI Katsuhiko  Ochanomizu University, Faculty of Science, Assistant Professor, 理学部, 助教授 (80222298)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsKleinian 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
Research Abstract

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.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (34 results)

All Other

All Publications (34 results)

  • [Publications] K.Matsuzaki: "Structural stability of Kleinian groups" Michigan Math.J.44. 21-36 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "The isomorphism theorem of Kleinian groups" Analysis and Topology. 507-513 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Conditional stability of Kleinian groups" Sci.Bull.Josai Univ.4. 25-28 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 松崎克彦: "クライン群の力学系--極限集合のハウスドルフ次元--" 日本数学会誌「数学」. 51. 掲載予定 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Hausdorff dimension of the limit sets of infinitely generated Kleinian groups" Math.Proc.Cambridge Phil.Soi.128 (To appear). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi and T.Harada: "On Teichmuller spaces of complex dynamics by entire functions" Bull.Hongkong Math.J.1. 257-266 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Matsuzaki and M.Taniguchi: "Hyperbolic manifold and Kleinian groups" Oxford Univ.Press, 253 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki and M.Taniguchi (他7名, 小島定吉監訳): "3次元幾何学とトポロジー(W.Thurston, Three-Dimensional Geometry and Topologyの日本語訳)" 培風館,

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Structural stability of Kleinian groups" Michigan Math.J.44. 21-36 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Conditional stability of Kleinian groups" Sci.Bull.Josai Univ.4-1. 25-28 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "The isomorphism theorem of Kleinian groups" Analysis and Topology. World Scientific. 507-513 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Matsuzaki: "Hausdorff dimension of the limit sets of infinitely generated Kleinian groups" Math.Proc.Cambridge Phil.Soi.128 (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Watanabe: "The initial-boundary value problems for the heat operator in non-cylindrical domains" J.Math.Soc.Japan. 49-3. 399-430 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Watanabe: "Double layer potentials of functions in a Besov space for a bounded domain with fractal boundary" Proceedings of the fifth international colloquim on finite or infinite dimensional complex analysis, Beijing. 337-343 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Watanabe: "Layer heat potentials for a bounded cyliner with fractal boundary" RIMS koukyuroku. 1042. 112-122 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Watanabe: "Besov spaces on fractal sets" Sci.Bull.Josai Univ. 5 (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi: "Logarithmic lifts of the family lambdaze^Z" RIMS koukyuroku. 988. 21-28 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi and T.Harada: "On Teichmuller spaces of complex dynamics by entire functions" Bull.Hingkong Math.J.1. 257-266 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi: "On topological completeness of decorated exponential famillies" Sci.Bull.Josai Univ.4. 1-10 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Maeda: "The quasi-Sazonov topology" RIMS koukyuroku. 1039. 81-94 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Matsuzaki and M.Taniguchi: Hyperbolic manifold and Kleinian groups. Oxford Univ.Press, (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Taniguchi: "On topological completeness of decorated exponential famillies" Sci.Bull.Josai Univ.4. 1-10 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Watanabe: "Besov spaces on fractal sets" Sci.Bull.Josai Univ.5 To appear. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Watanabe: "Layer heat potentials for a bounded cylinder with fractal boundary" 数理解析研究所講究録. 1042. 112-122 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Maeda: "The quasi-Sazonov topology" 数理解析研究所講究録. 1039. 81-94 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Matsuzaki and M.Taniguchi: "Hyperbolic manifold and Kleinian groups" Oxford Univ.Press, 253 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Matsuzaki and M.Taniguchi(他7名,小島定吉監訳): "3次元幾何学とトポロジー(W.Thurston,Three-Dimensional Geometry and Topologyの日本語訳)" 培風館, (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Matsuzaki: "Structural stability of Kleinian groups" Michigan Math.J.44. 21-36 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 松崎克彦: "クライン群の極限集合のハウスドルフ次元" 「ハウスドルフ次元-計算へのアプローチ」研究集会報告集. 132-138 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Taniguchi and T.Harada: "On Teichmuller spaces of complex dynamics by entire functions" Bull.HongKong Math.J.1. 257-266 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 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)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.Watanabe: "The initial-boundary value problems for the heat operator in noncylindrical domains" J.Math.Soc.Japan. 49・3. 399-430 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 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).

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Matsuzaki & M.Taniguchi: "Oxford Univ.Press" Hyperbolic manifolds and Kleinian groups, 272 (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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