On the characterization of measure-expansive differentiable maps

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05167
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Utsunomiya University
• Principal Investigator: Sakai Kazuhiro 宇都宮大学, 教育学部, 教授 (30205702)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 拡大性 / 確率測度 / 測度拡大性 / 双曲性

Outline of Final Research Achievements

The purpose is to characterize the dynamics of differentiable maps with expansive measures from the viewpoint of geometric theory of dynamical systems, but not completed yet. It has shown that the C1-interior of the set of regular differentiable maps which are measure expansive for any measure coincides with quasi-Anosov systems, and that any differentiable map in the C1-interior of the set of positively measure expansive maps for any measure is hyperbolic when the set of periodic points of the system is composed of infinitely many expanding periodic points and finitely many contracting periodic points.
In the above process, we introduced the notion of shadowable measures from the measure theoretical view point. Then, the usefulness of the notion has been shown by obtaining a characterization of the dynamical system with shadowable measures, and, by estimating the measures of shadowable pseudo-orbits for the systems without the shadowing property.

Free Research Field

力学系理論

Academic Significance and Societal Importance of the Research Achievements

本研究は，測度論との融合という新たな視点から特異点の存在を踏まえた微分可能力学系における一様双曲系の分類（特徴付け）を行おうというものであり，単に分類（特徴付け）のための新たな視点・概念の導入にとどまることなく，本研究の推進過程で発見・開発される新たな解析手法により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る。
また，測度拡大的の概念はカオスの定義の構成要素である初期値鋭敏性とも深い関係がある。本研究では，その概念を測度論的な視点，すなわち「観測可能の視点」から研究を展開しようとするもので，そこで得られた研究成果や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。