2020 Fiscal Year Final Research Report
Study of mean dimension of dynamical systems
| Project/Area Number |
18K03275
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Kyushu University (2019-2020)
Kyoto University (2018) |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2021-03-31
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| Keywords | 力学系 / エルゴード理論 / 平均次元 / レート歪み理論 |
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| Outline of Final Research Achievements |
The purpose of this project to to study mean dimension. Mean dimension is the number of degrees of freedom per unit time for describing dynamical systems. We found the ``double variational principle for mean dimension''. Given a minimal dynamical system, we proved that its mean dimension is equal to the minimax value of the rate distortion dimension (the quantity in information theory). This discovery opens a very new interaction between dynamical systems theory and information theory.
We also found several interesting things around mean dimension. In particular, we solved a problem about ``marker property'', which asks whether there is a free dynamical system violating the marker property.
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| Free Research Field |
数学
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| Academic Significance and Societal Importance of the Research Achievements |
力学系の研究では，伝統的に情報理論との関連が重要視されており，例えば力学系のエントロピーの理論では，「位相的エントロピー」という量を「測度論的エントロピー」という情報理論的量を用いて表す「変分原理」が有名である．類似の理論を「平均次元」に対してつくることは，平均次元が２０年前に定義されて以来自然な問題だったが，これまで進展がなかった．今回，それをついに成し遂げることができた．
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