Ergodicity of volume preserving dynamical systems and its applications
| Project/Area Number |
15K17550
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | エルゴード性 / 葉層構造 / 横断性 / 葉層 |
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| Outline of Final Research Achievements |
We studied geometric criteria for the ergodicity of volume preserving dynamical systems. It is widely known that the Hopf argument is a simple but strong method in the ergodic theory of Anosov systems. For a broader class of dynamical systems, called the non-uniformly hyperbolic systems, we extend the method under an assumption on the dimension of the phase space or the associated foliations. As an application, we showed that the ergodicity on every closed three-manifold follows from a topological property of the system. Further, it has turned out that every topological transitive topological Anosov system is ergodic.
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| Academic Significance and Societal Importance of the Research Achievements |
力学系のエルゴード理論における基本的かつ重要な問題の一つに，与えられた力学系のエルゴード性を判定する問題がある．この問題は，数論的変換が定める力学系などの特殊な場合を除けば，一般に極めて難しい問題である．ところでAnosov型とよばれる可微分力学系については，Hopf議論とよばれる手法により，いわば幾何学的にエルゴード性を示すことができる．本研究ではこの手法を，Anosov型力学系より広範な，そして十分に一般的な力学系について拡張することを目指し，次元に関する仮定の下でこれを実現した．
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Report
(5 results)
Research Products
(13 results)
