1998 Fiscal Year Final Research Report Summary
A topological study of generalized dynamical systems
| Project/Area Number |
09640090
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | CHIBA UNIVERSITY |
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Principal Investigator |
INABA Takashi Faculty of Science, Professor, 理学部, 教授 (40125901)
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| 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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | generalized dynamical systems / foliation / pseudo-orbit / pseudoleaf / pseudoleaf tracing property / expansivity / stability / entropy |
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| 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.
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