1999 Fiscal Year Final Research Report Summary
Synthetic Study of Foliations and Dynamical Systems
| Project/Area Number |
09440042
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Nihon University |
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Principal Investigator |
MATUMOTO Shigenori Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (80060143)
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| Co-Investigator(Kenkyū-buntansha) |
KOJIMA Sadayoshi Tokyo Institute of Technology, Graduate school of Information Science and Engineering, 大学院・情報理工学研究科, 教授 (90117705)
TSUBOI Takashi University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院・数理科学研究科, 教授 (40114566)
INABA Takashi Chiba University, Graduate School of Science and Technology, Professor, 大学院・自然科学研究科, 教授 (40125901)
MINAKAWA Hiroyuki Hirosaki University, Faculty of Education, Assistant Professor, 教育学部, 助教授 (30241300)
NAKAYAMA Hiromichi Hiroshima University, Faculty of Integarated Arts and Sciences, Lecturer, 総合科学部, 講師 (30227970)
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| Project Period (FY) |
1997 – 1999
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| Keywords | foliations / area preserving maps / rotation number / rotation vector / symplectic diffeomorphism / Hamiltonian flow / holonomy group / transverse measure |
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| Research Abstract |
Dynamical Systems without Minimal Sets There exists a flow without minimal set on a certain surface of infinite type. Also there is a diffeomorphism without minimal set on an open annulus.
Transverse Intersection of Two Foliations on 3-manifolds The stable and unstable foliations of the suspension Anosov flow are shown to intersect transversely in an unique way. On the countrary, there are nonstandard way of intersection of the stable and unstable foliations of the geodesic flows of hyperbolic surfaces.
Flows on 3-manifolds Consider a nonsigular vector field X on a 3-manifold M and the flow φ defined by X. The differential of the flow defines a flow ψ on the projectived normal bundle P of the flow X. Thus the invariant measure ν of ψ on P gives rise to an invariant measure μ of φ on M. We showed that one of the followings occurs. (1) For μ-a.e. x ∈ M, Supp(ν) ∩ PィイD2xィエD2 is one point. (2) For μ-a.e. x ∈ M, Supp(ν) ∩ PィイD2xィエD2 consists of two points. (3) For μ-a.e. x ∈ M. PィイD2xィエD2 is contained in Supp(ν).
We also showed that for the lift of ψ to the cyclic convering of P which unfolds the fibers, one of the followings holds. (1) All the orbits are bounded. (2) All the orbits are proper. (3) There is a dense orbits.
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