The characterizations of dynamical systems using shadowable measures

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03578
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Utsunomiya 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
• 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

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.

Free Research Field

力学系理論

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。