2000 Fiscal Year Final Research Report Summary
A study of minimal sets in differentiable flows and foliations
| Project/Area Number |
11640062
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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 Graduate School of Science and Technology, Prof., 大学院・自然科学研究科, 教授 (40125901)
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| Co-Investigator(Kenkyū-buntansha) |
HINO Yoshiyuki Faculty of Science, Prof., 理学部, 教授 (70004405)
KUGA Ken'ichi Faculty of Science, Associate .Prof., 理学部, 助教授 (30186374)
NAKAYAMA Hiromichi Hiroshima Univ., Faculty of Integrated Arts and Sciences, Associate Prof., 総合科学部, 助教授 (30227970)
SUGIYAMA Ken-ichi Faculty of Science, Associate Prof., 理学部, 助教授 (90206441)
TAKAGI Ryoichi Faculty of Science, Prof., 理学部, 教授 (00015562)
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| Project Period (FY) |
1999 – 2000
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| Keywords | flow / invariant fiber measure / Ruelle invariant / amenable group |
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| Research Abstract |
In this research we made measure theoretic and topological investigation of differentiable flows on 3-dimensional manifolds. Through an effort to understand more concretely Zimmer's ergodic theory on group actions, we established a basis of finding new aspects of dynamics of flows. Our method is as follows : Given a flow we lift it to the total space of its projectivized normal bundle. Then, by disintegrating an invariant measure on the total space, we obtain a measure on each fiber. We Call the family of measures thus obtained an invariant fiber measure. An invariant fiber measure provides us with a setup for measuring the twisting phenomenon of the flow.
In this research we established some fundamental theorems on invariant fiber measures, giving careful and accessible proofs for all of them. We also obtained a new representation formula of the Ruelle invariant by making use of an invariant fiber measure, Moreover, as an application of this formula, we showed the following result : Consider the group of measure preserving diffeomorphisms of the 2-dimensional disk and the Ruelle map defined on this group. Then the map becomes a homomorphism whenever we restrict it to an amenable subgroup.
We gave a one-hour talk on this result at an international symposium on foliations held at Warsaw in 2000.
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Research Products
(20 results)
