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Study of mean dimension of dynamical systems

Research Project

Project/Area Number 18K03275
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionKyushu University (2019-2020)
Kyoto University (2018)

Principal Investigator

Tsukamoto Masaki  九州大学, 数理学研究院, 教授 (70527879)

Project Period (FY) 2018-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords力学系 / エルゴード理論 / 平均次元 / レート歪み理論 / 情報理論 / マーカー性質 / 周期点 / 位相的エントロピー / 幾何解析
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.

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.

Report

(4 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (10 results)

All 2021 2020 2019 2018 Other

All Int'l Joint Research (3 results) Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results) Presentation (4 results) (of which Invited: 4 results)

  • [Int'l Joint Research] Polish Academy of Sciences(ポーランド)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Hebrew University(イスラエル)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Hebrew University(イスラエル)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Symbolic dynamics in mean dimension theory2020

    • Author(s)
      Mao Shinoda, Masaki Tsukamoto
    • Journal Title

      Ergodic Theory and Dynamical Systems

      Volume: - Issue: 8 Pages: 2542-2560

    • DOI

      10.1017/etds.2020.47

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Double variational principle for mean dimension with potential2020

    • Author(s)
      Masaki Tsukamoto
    • Journal Title

      Advances in Mathematics

      Volume: 361 Pages: 106935-106935

    • DOI

      10.1016/j.aim.2019.106935

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Double variational principle for mean dimension2019

    • Author(s)
      Elon Lindenstrauss, Masaki Tsukamoto
    • Journal Title

      Geometric and Functional Analysis

      Volume: 29 Issue: 4 Pages: 1048-1109

    • DOI

      10.1007/s00039-019-00501-8

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 平均次元入門2021

    • Author(s)
      塚本真輝
    • Organizer
      日本数学会年会 総合講演
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] フルシフトの平均次元と関連する諸問題2020

    • Author(s)
      塚本真輝
    • Organizer
      一般位相幾何学とその関連分野の発展
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] 力学系の平均次元と情報理論2019

    • Author(s)
      塚本真輝
    • Organizer
      日本数学会 幾何学賞受賞特別講演
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 平均次元に対する二重変分原理2018

    • Author(s)
      塚本真輝
    • Organizer
      幾何学シンポジウム 基調講演
    • Related Report
      2018 Research-status Report
    • Invited

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Published: 2018-04-23   Modified: 2022-01-27  

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