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1998 Fiscal Year Final Research Report Summary

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
Keywordsgeneralized dynamical systems / foliation / pseudo-orbit / pseudoleaf / pseudoleaf tracing property / expansivity / stability / entropy
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.

  • Research Products

    (17 results)

All Other

All Publications (17 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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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

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Published: 1999-12-08  

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