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Study of dynamical and geometrical properties of maps on separable metric spaces

Research Project

Project/Area Number 19540063
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

KATO Hisao  University of Tsukuba, 大学院・数理物質科学研究科, 教授 (70152733)

Co-Investigator(Kenkyū-buntansha) SAKAI Katsuro  筑波大学, 大学院・数理物質科学研究科, 准教授 (50036084)
KAWAMURA Kazuhiro  筑波大学, 大学院・数理物質科学研究科, 准教授 (40204771)
KANETO Takeshi  筑波大学, 大学院・数理物質科学研究科, 講師 (70107340)
YAMAZAKI Kaori  高崎経済大学, 経済学部, 准教授 (80301076)
MINAMI Nariyuki  慶応義塾大学, 医学部, 教授 (10183964)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywordsカオス / エントロピー / 拡大写像 / 連続体 / 非分解空間 / フラクタル / 拡大同相写像 / エルゴート理論 / 写像空間 / 非分解空聞
Research Abstract

We studied chaotic properties of maps of compact metric spaces by use of topological theory and ergodic theory. In this study, we proved that hereditarily indecomposable continua admit no expansive homeomorphisms. Also by use of Alexandroff-Urysohn metrization theorem we obtained very strong theorems concerning ratios of expanding maps and fractal dimensions. These results are very important in the theories of geometric topology and dynamical systems.

Report

(4 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (23 results)

All 2010 2009 2008 2007

All Journal Article (15 results) (of which Peer Reviewed: 13 results) Presentation (8 results)

  • [Journal Article] Fractal Metrics of Ruelle Expanding Maps and Expanding Ratios2010

    • Author(s)
      H. Kato, C. Fujita, M. Matsumoto
    • Journal Title

      Topology and its Applications 157

      Pages: 615-628

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Fractal Metrics of Ruelle Expanding Maps and Expanding Ratios2010

    • Author(s)
      C.Fujita, H.Kato, M.Matsumoto
    • Journal Title

      Topology and its Applications 157

      Pages: 615-628

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Almost periodic points and minimal sets in topological spaces2009

    • Author(s)
      H. Kato, C. Fujita
    • Journal Title

      Applied General Topology 10

      Pages: 239-244

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Almost periodic points and minimal sets in topological spaces2009

    • Author(s)
      C.Fujita, H.Kato
    • Journal Title

      Applied General Topology 10

      Pages: 239-244

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hereditarily indecomposable compacta do not admit expansive homeomorphisms2008

    • Author(s)
      H. Kato, C. Mouron
    • Journal Title

      Proc.Amer.Math.Soc. 136

      Pages: 3689-3696

    • NAID

      120007131232

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hausdorff hyperspaces of R^m and their dense subspaces2008

    • Author(s)
      Kubis, Wieslaw, K. Sakai
    • Journal Title

      J.Math.Soc.Japan 60, no. 1

      Pages: 193-217

    • NAID

      10024331526

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hereditarily indecomposable compacta do not admit expansive homeomorphisms2008

    • Author(s)
      Hisao Kato and C. Mouron
    • Journal Title

      Proc. Amer. Math. Soc. 136

      Pages: 3689-3696

    • NAID

      120007131232

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Expanding Ratios, box-counting dimension and Hausdorff dimension,2008

    • Author(s)
      Hisao Kato
    • Journal Title

      数理解析研究所講究録 1578

      Pages: 28-35

    • Related Report
      2008 Annual Research Report
  • [Journal Article] The nonexistence of expansive homeomorphisms on hereditarily indecomposable compacta2008

    • Author(s)
      Hisao Kato
    • Journal Title

      数理解析研究所講究録 1578

      Pages: 36-39

    • Related Report
      2008 Annual Research Report
  • [Journal Article] Hereditarily indecomposable compacta do not admit expansive homeomo rphisms2008

    • Author(s)
      Hisao Kato and C. Mouron
    • Journal Title

      Proc. AMS (印刷中)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher Dimensional Bruckner-Garg Type Theorem2007

    • Author(s)
      H. Kato
    • Journal Title

      Topology and its Applications 154

      Pages: 1690-1702

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Topological Entropy of Maps on Regular Curves2007

    • Author(s)
      H. Kato
    • Journal Title

      Topology and its Applications 154

      Pages: 1027-1031

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Root closed function algebras on compacta of large dimension2007

    • Author(s)
      Brodskiy, N., Dydak, J., Karasev, A., K. Kawamura
    • Journal Title

      Proc.Amer.Math.Soc. 135, no.2

      Pages: 587-596

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Higher Dimensional Bruckner-Garg Type Theorem2007

    • Author(s)
      Hisao Kato
    • Journal Title

      Topology and its Applications 154

      Pages: 1690-1702

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Topological Entropy of Maps on Regular Curves2007

    • Author(s)
      Hisao Kato
    • Journal Title

      Topology and its Applications 154

      Pages: 1027-1031

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Chaotic continua of continuum-wise expansive homeomorphisms2009

    • Author(s)
      加藤久男
    • Organizer
      「一般位相幾何学及び幾何学トポロジーに関する研究」数理解析研究所研究集会
    • Place of Presentation
      京都大学
    • Year and Date
      2009-10-16
    • Related Report
      2009 Final Research Report
  • [Presentation] Chaotic continua of continuum-wise expansive homeomorphisms2009

    • Author(s)
      H.Kato
    • Organizer
      数理解析研究所研究集会
    • Place of Presentation
      京都大学
    • Year and Date
      2009-10-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] Dynamical systems which realize given random bi-sequences of points on their orbits2009

    • Author(s)
      加藤久男
    • Organizer
      第44回位相空間論シンポジウム
    • Place of Presentation
      島根大学
    • Year and Date
      2009-05-30
    • Related Report
      2009 Final Research Report
  • [Presentation] Dynamical systems which realize given random bi-sequences of points on their orbits2009

    • Author(s)
      H.Kato
    • Organizer
      位相空間論シンポジウム
    • Place of Presentation
      島根大学
    • Year and Date
      2009-05-30
    • Related Report
      2009 Annual Research Report
  • [Presentation] Expanding Ratios, box-counting dimension and Hausdorff dimension2008

    • Author(s)
      加藤久男
    • Organizer
      「一般位相幾何学及び幾何学トポロジーとその応用」数理解析研究所研究集会
    • Place of Presentation
      京都大学
    • Year and Date
      2008-10-08
    • Related Report
      2009 Final Research Report
  • [Presentation] The nonexistence of expansive homeomorphisms on hereditarily indecomposable compacta2007

    • Author(s)
      加藤久男
    • Organizer
      「一般位相幾何学及び幾何学トポロジー」数理解析研究所研究集会
    • Place of Presentation
      京都大学
    • Year and Date
      2007-10-19
    • Related Report
      2009 Final Research Report
  • [Presentation] 高次元コンパクト距離空間より多様体への連続写像が作る写像空間について2007

    • Author(s)
      加藤久男
    • Organizer
      日本数学会一般講演
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2009 Final Research Report
  • [Presentation] 高次元コンパクト距離空間より多様体への連続写像が作る写像空間について2007

    • Author(s)
      加藤久男
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2007 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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