Study of chaotic dynamical systems by use of geometric topology

• Project/Area Number: 22540065
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: University of Tsukuba
• Principal Investigator: KATO Hisao 筑波大学, 数理物質系, 教授 (70152733)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: colorings / 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 / 位相力学系 / フラクタル次元 / 幾何学的トポロジー / 彩色問題 / 拡大同相写像 / 位相次元 / box次元 / coloring / 拡大定数 / flow

Research Abstract

We studied eventual colorings of maps. We obtained some important theorems of eventual coloring numbers. By use of Alexandroff-Urysohn metrization theorem we obtained very strong theorems concerning topological dimension and box-counting dimension. Also, we studied chaotic measure-preserving dynamical systems of compact manifolds. These results are very important in the theories of geometric topology and dynamical systems.

Research Products

• [Journal Article] Periodic points, compactificationsand eventual colorings of maps (2013)
  • Author(s): H. Kato
  • Journal Title: Topology Appl. Volume: 160 Pages: 685-691
  • Peer Reviewed
• [Journal Article] Dynamical properties of finiteDimensional metric spaces and zerodimensional covers (2013)
  • Author(s): Y. Ikegami, H. Kato and A. Ueda
  • Journal Title: Topology and its Applications Volume: 160 Pages: 564-574
  • Peer Reviewed
• [Journal Article] Eventual colorings of homeomorphisms (2013)
  • Author(s): Y. Ikegami, H. Kato and A. Ueda
  • Journal Title: J. Math. Soc. Japan Volume: 65, No 2 Pages: 375-387
  • NAID: 10031177285
  • Peer Reviewed
• [Journal Article] Periodic points, compactifications and eventual colorings of maps (2013)
  • Author(s): H. Kato
  • Journal Title: Topology Appl. Volume: 160 Pages: 685-691
  • Peer Reviewed
• [Journal Article] Dynamical properties of finite-dimensional metric spaces and zero-dimensional covers (2013)
  • Author(s): Y. Ikegami, H. Kato and A. Ueda
  • Journal Title: Topology Appl. Volume: 160 Pages: 564-574
  • Peer Reviewed
• [Journal Article] Eventual colorings of homeomorphisms (2013)
  • Author(s): Y. Ikegami, H. Kato and A. Ueda
  • Journal Title: J. Math. Soc. Japan Volume: 65 Pages: 375-387
  • NAID: 10031177285
  • Peer Reviewed
• [Journal Article] Addendum to: Characterizations of topological dimension by use of normal Sequences of finite open covers and Pontrjagin-Shnirelmann' theorem (2011)
  • Author(s): H. Kato
  • Journal Title: J. Math. Soc. Japan Volume: 63 Pages: 977-983
  • Peer Reviewed
• [Journal Article] Characterizations of topological dimension by use of normal Sequences of finite open covers and Pontrjagin-Shnirelmann' theorem (2011)
  • Author(s): H. Kato and M. Matsumoto
  • Journal Title: J. Math. Soc. Japan Volume: 63 Pages: 919-976
  • Peer Reviewed
• [Journal Article] Dynamical systems which realize given random bi-sequences of points on their orbits (2011)
  • Author(s): H. Kato
  • Journal Title: Topology and its Applications Volume: 158 Pages: 1163-1171
  • NAID: 120007130874
  • Peer Reviewed
• [Journal Article] On indecomposability and composants of chaotic continua II (2011)
  • Author(s): H. Kato
  • Journal Title: Topology and its Applications Volume: 158 Pages: 647-652
  • Peer Reviewed
• [Journal Article] Addendum to: Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem (2011)
  • Author(s): H.Kato
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 63 Issue: 3 Pages: 977-983
  • DOI: 10.2969/jmsj/06330977
  • NAID: 10029321974
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem (2011)
  • Author(s): H.Kato, M.Matsumoto
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 63 Issue: 3 Pages: 919-976
  • DOI: 10.2969/jmsj/06330919
  • NAID: 10029321958
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Fractal Metrics of Ruelle Expanding Maps and Expanding Ratios (2010)
  • Author(s): C. Fujita, H. Kato and M. Matsumoto
  • Journal Title: Topology and its Applications Volume: 157 Pages: 615-628
  • Peer Reviewed
• [Journal Article] On eventual coloring numbers
  • Author(s): Y. Ikegami, H. Kato and A. Ueda
  • Journal Title: Topology Proceedings
  • Peer Reviewed
• [Presentation] Compactifications of a space and periodic points of maps (2012)
  • Author(s): 加藤久男
  • Organizer: 2012ジェネラルトポロジーシンポジウム
  • Place of Presentation: 神戸大学
  • Year and Date: 2012-12-13
• [Presentation] Zero-dimensional covers of dynamcal systems (2012)
  • Author(s): 加藤久男
  • Organizer: 南京大学大学での招待講演
  • Place of Presentation: 中国
  • Year and Date: 2012-09-26
• [Presentation] Eventual colorings of maps of separable metric spaces (2012)
  • Author(s): 加藤久男
  • Organizer: International Conference on Topology and the Related Fields
  • Place of Presentation: 南京師範大学、中国
  • Year and Date: 2012-09-24
• [Presentation] Eventual colorings of maps of separable metric spaces (2012)
  • Author(s): H. Kato
  • Organizer: International Conference on Topology and the Related Fields
  • Place of Presentation: 南京師範大学（中国）
  • Invited
• [Presentation] Zero-dimensional covers of dynamcal systems (2012)
  • Author(s): H. Kato
  • Organizer: 南京大学数学教室
  • Place of Presentation: 南京大学（中国）
  • Invited
• [Presentation] Dynamical properties of homeomorphisms and zero-dimensional covers (2011)
  • Author(s): 加藤久男
  • Organizer: 2011ジェネラルトポロジーシンポジウム
  • Place of Presentation: 埼玉大学
  • Year and Date: 2011-12-20
• [Presentation] Colorings and eventual colorings, 一般位及び幾何学的的トポロジーとその応用 (2011)
  • Author(s): 加藤久男
  • Organizer: 京都大学数理解析研究所研究集会
  • Year and Date: 2011-10-19
• [Presentation] Eventual Colorings of homeomorphisms of separable metric spaces (2011)
  • Author(s): 加藤久男
  • Organizer: 京都大学数理解析研究所研究集会
  • Place of Presentation: 京都大学数理解析研究所(京都市)(招待講演)
  • Year and Date: 2011-10-19
• [Presentation] 距離の構成による色々な次元, 一般位相幾何学及び幾何学的的トポロジーの最近の話題とその応用 (2010)
  • Author(s): 加藤久男
  • Organizer: 京都大学数理解析研究所研究集会
  • Year and Date: 2010-10-13
• [Presentation] 距離の構成による色々な次元 (2010)
  • Author(s): H.Kato
  • Organizer: 京都大学数理解析研究所研究集会
  • Place of Presentation: 京都大学
  • Year and Date: 2010-10-13
• [Presentation] Characterizations of topological dimension by use of normal Sequences of finite open covers and Pontrjagin-Shnirelmann' theorem (2010)
  • Author(s): H.Kato
  • Organizer: 日本-メキシコトポロジー国際会議
  • Place of Presentation: University of Colima, Mexico 招待講演
  • Year and Date: 2010-09-29
• [Presentation] Characterizations of topological dimension by use of normal Sequences of finite open covers and Pontrjagin-Shnirelmann' theorem (2010)
  • Author(s): 加藤久男
  • Organizer: Jamex(日本-メキシコ トポロジー国際会議)
  • Place of Presentation: University of Colima, Mexico
  • Year and Date: 2010-09-28