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The characterizations of dynamical systems using shadowable measures

Research Project

Project/Area Number 19K03578
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionUtsunomiya University

Principal Investigator

Sakai Kazuhiro  宇都宮大学, 共同教育学部, 教授 (30205702)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2021: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2020: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords擬軌道尾行性 / 尾行可能測度 / 確率測度 / 一様双曲型 / 非一様双曲型 / 占有的分解 / エルゴード的測度 / 力学系理論 / 双曲性
Outline of Research at the Start

fを多様体M上の微分同相写像とし, M上の確率測度全体を\cal{M}(M)とする。また、f-不変確率測度全体を\cal{M}_f(M)で、エルゴード的f-不変確率測度全体を\cal{M}_f~e(M)で表す。本研究では、\cal{PS}={f:\mu-尾行可能 (\forall \mu \in \cal{M}(M))}, \cal{IS}={f:\mu-尾行可能 (\forall \mu \in \cal{M}_f(M))}、そして \cal{ES}={f:\mu-尾行可能 (\forall \mu \in \cal{M}_f^e(M))}を考察対象とし、これらの集合のC~1-位相に関する内点を一様双曲性や非一様双曲性の概念を用いて特徴付ける。

Outline of Final Research Achievements

We aimed for the new development of the shadowing theory of dynamical systems by generalizing the notion of shadowing property from the viewpoint of measure theory and extending our dynamical systems to non-uniformly hyperbolic systems, but not completed yet. In this project, we introduced the notion of shadowable measures for the set of pseudo-orbits, and try to characterize the diffeomorphisms admitting the shadowable measures by analyzing the behavior of shadowable pseudo-orbits from measure-theoretical viewpoint.
We have two remarkable results by making use of the shadowable measures. One is for a system which does not satisfy the shadowing property in general, we obtained the quantitative estimation for the set of pseudo-orbits of the system by the shadowable measure. The other is in the context of C^1 diffeomorphisms, we proved that there is a C^1 open set of diffeomorphisms such that for any element of the set, the Lebesgue measure itself is shadowable for the element.

Academic Significance and Societal Importance of the Research Achievements

本研究では測度論の視点から擬軌道尾行性の概念を一般化し,研究対象とする力学系を非一様双曲系に拡張することで尾行性理論の新展開を目指した。その完成には至らなかったが,擬軌道集合の尾行可能測度による量的評価や,多様体上のルベーグ測度が尾行可能測度となる力学系の開集合の存在など,単に力学系の特徴付け研究にとどまることなく,研究の推進過程で発見された新たな解析手法や知見など,力学系理論全体における研究の進展に貢献することができた。
また,尾行可能測度の概念は非一様双曲系に適用可能であり,カオス研究とも深い関係がある。本研究で得られた新たな知見はカオス理論研究の応用面においても大きな寄与が期待できる。

Report

(5 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (4 results)

All 2021 2020 2019

All Journal Article (4 results) (of which Peer Reviewed: 4 results)

  • [Journal Article] Positively expansive maps and the limit shadowing properties2021

    • Author(s)
      Kazuhiro Sakai
    • Journal Title

      J. Korean Math. Soc.

      Volume: 58 Pages: 207-218

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures2021

    • Author(s)
      Sakai Kazuhiro、Sumi Naoya
    • Journal Title

      Axioms

      Volume: 10 Issue: 1 Pages: 38-38

    • DOI

      10.3390/axioms10010038

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Shadowing Property and Invariant Measures Having Full Supports2020

    • Author(s)
      K. Moriyasu, K. Sakai and N. Sumi
    • Journal Title

      Qualitative Theory of Dynamical Systems

      Volume: - Issue: 1 Pages: 1-7

    • DOI

      10.1007/s12346-020-00338-9

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] The measures of shadowable pseudo-orbits2019

    • Author(s)
      K. Sakai and N. Sumi
    • Journal Title

      Dynamical Systems: An International Journal

      Volume: - Issue: 3 Pages: 369-381

    • DOI

      10.1080/14689367.2019.1703907

    • Related Report
      2019 Research-status Report
    • Peer Reviewed

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Published: 2019-04-18   Modified: 2024-01-30  

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