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A topological study of generalized dynamical systems

Research Project

Project/Area Number 09640090
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionCHIBA UNIVERSITY

Principal Investigator

INABA Takashi  Faculty of Science, Professor, 理学部, 教授 (40125901)

Co-Investigator(Kenkyū-buntansha) KOSHIKAWA Hiroaki  Faculty of Education, Associate Professor, 教育学部, 助教授 (60000866)
SUGIYAMA Ken-ichi  Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
TAKAGI Ryoichi  Faculty of Science, Professor, 理学部, 教授 (00015562)
HINO Yoshityki  Faculty of Science, Professor, 理学部, 教授 (70004405)
KUGA Ken'ichi  Faculty of Science, Associate Professor, 理学部, 助教授 (30186374)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1998: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1997: ¥1,600,000 (Direct Cost: ¥1,600,000)
Keywordsgeneralized dynamical systems / foliation / pseudo-orbit / pseudoleaf / pseudoleaf tracing property / expansivity / stability / entropy / 位相安定性
Research Abstract

By generalized dynamical systems, we mean a wide class of systems including classical dynamical systems, foliations, pseudogroups and relations. In this research we tried to extend various notions and theorems in classical dynamical systems to generalized systems. The results obtained are as follows.
1. The Reeb foliation and Anosov foliations have the pseudoleaf tracing property.
2. Expansive foliations with the pseudoleaf tracing property are semi-stable (This is a generaliza- tion of Bowen-Walters theorem in classical dynamical systems).
3. A group action has the pseudo-orbit tracing property if and only if its suspension foliation has the pseudoleaf tracing property.
4. A group action has the semi-stability or the expansivity if and only if its suspension foliation has the same property.
5. We recognized that the notion of the center of mass (defined by Cheeger) is useful in the construction of a pseudoleaf from a given countable subset in the ambient manifold.
6. It seems that the growth of the number of compact leaves cannot be estimated from above by the geometric entropy. We are now trying to produce a counterexample.
7, In the case of foliations it seems that the stability does not imply the pseudoleaf tracing property.
These results will be published in Tokyo J.Math. A related result has been published in Erg. Th. Dyn. Sys.
We note that the notion of the pseudoleaf (which was introduced in this research) has been applied by Walczak to the new definition of the geometric entropy and the existence of virtual leaves.
Hino investigated the stability of processes. Koshikawa studied the cut-and-pasting method of group actions. And Takagi studied the geometry of real hypersurfaces in the complex projective spaces.

Report

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

    (29 results)

All Other

All Publications (29 results)

  • [Publications] Takashi Inaba: "Expansivity, psendsleaf tracing proparty and semi stability of foliations" Tokyo Journal of Mathematics. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takashi Inaba: "An example of a flow on a non-compact surface without minimal set" Ergodic Theory & Dynamical Systems. 19-1. 31-33 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshiyuki Hino & Satoru Murakami: "Skew product flows of quasi-processes and stabilities" Fields Institute Communications. 21. 269-277 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] I-B, Kim, B. H. Kim & Ryoichi Takagi: "The rigidity for real hypersurfaces in a complex projective space" Tohoku Mathematical Journal. 50-4. 531-536 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y. J. Shin and Ryoichi Takagi: "On almost parallel submanifolds in S^<2n+1>" Kyungpook Mathematical Journal. 37-1. 175-180 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Tamio Hara & Hiroaki Koshikawa: "A homerorphism between an equivariant SK ring and the Burnside ring for U_4" Hokkaido Mathematical Journal. 28-3(印刷中). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takashi Inaba: "Expansivity, pseudoleaf tracing property and semistability of foliations" Tokyo J.Math.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takashi Inaba: "An example of a flow on a non-compact surface which has no minimal set" Erg.Th.Dyn.Sys.19-1. 31-33 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshiyuki Hino and Satoru Murakami: "Skew product flows of quasi-processes and stabilities" Fields Inst.Commun.21. 269-277 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshiyuki Hino and Satoru Murakami: "A generalization of processes and stabilities in abstract functional differential equations" Funkc.Ekvac.41-2. 235-255 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshiyuki Hino and Satoru Murakami: "Almost periodic processes and the existence of almost periodic solutions" Electron.J.Qual.Th.Diff.Eq.1-3. 1-19 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kurihara and Ryoichi Takagi: "A note on the type number of real hypersurfaces in" Tsukuba J.Math.22-3(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] I-B.kim, B.H.Kim and Ryoichi Takagi: "The rigidity for real hypersurfaces in a complex projective space" Tohoku Math.J.50-4. 531-536 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.J.Shin and Ryoichi Takagi: "On almost parallel submanifolds in S^<2n+1>" Kyungpook Math.J.37-1. 175-180 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Sadahiro Maeda and Ryoichi Takagi: "Circular geodesic submanifolds with parallel mean curvature vector in a non-flat complex space form" Proc.Japan Acad., Ser.A. 73-4. 55-57 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Tamio Hara and Hiroaki Koshikawa: "A homomorphism between an equivariant SK ring and the Burnside ring for Z_4" Hokkaido Math.J.28-3(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Tamio Hara and Hiroaki Koshikawa: "Cutting and pasting of G manifolds with boundary" Kyushu J.Math.51-1. 165-178 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takashi Inaba: "An example of a flow on a non-campact surface without minimal set" Ergodic Theory & Dynamical Systems. 19(発表予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Yoshiyuki Hino & Satoru Murakami: "Almost periodic processes and the existence of almost periodic solutions" EJQTDE. 1・3. 1-19 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Kurihara & Ryoichi Takagi: "A note on the type number of real hypersurfaces in Pn(C)" Tsukuba J.Math.22・3(発表予定). (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] I-B.Kim,B-H.Kim & Ryoichi Takagi: "The rigidity for real hypersurfaces in a conplex projedive space" Tohoku Math.J.50・4. 531-536 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Tamio Hara & Hiroaki Koshikawa: "A homomorphism between an equivariant SK ring and the Buruside ring for Z_4" Hokkaido Math.J.28・3(発表予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Yoshiyuki Hino & Satoru Murakami: "Skow product flows of quasi-processes and stability" Fields Institute Communications. 21. 269-277 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Takashi Inada: "An example of a flow on a noncompact surface without minimal set" Ergodic Theory&Dynamical Systems. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Yoshiyuki Hino&Satoru Murakami: "A generalization of processes and stabilities in abstract functional differential equations" Funkcialaj Ekvacioj. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Yoshiyuki Hino&Satoru Murakami: "Skew product flows of quasi-processes and stabilities" The Fields Institute Communications Series. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Sadahiro Maeda&Ryoichi Takagi: "Circular geodesic subwanifolds with parallel mean cuwature vector in a non-flat complex space form" Proceedings of the Japan Academy. 73・4. 55-57 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Yang Jae Shin&Ryoichi Takagi: "On almost parallel subwanifolds in S^<2n+1>" Kyungpook Mathematical Journal. 37・1. 175-180 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Tamio Hara&Hiroaki Koshikawa: "Cutting and pasting of G-manifolds with boundary" Kyushuu Journal of Mathematics. 51・1. 165-178 (1997)

    • Related Report
      1997 Annual Research Report

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

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