wild behavior of partially hyperbolic dynamics and its smoothness

• Project/Area Number: 18K03276
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Doshisha University (2020-2022); Kyoto University (2018-2019)
• Principal Investigator: Asaoka Masayuki 同志社大学, 理工学部, 教授 (10314832)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 部分双曲力学系 / 分岐理論 / 微分可能力学系 / 双曲力学系 / 力学系理論 / 葉層構造 / 力学系 / カントール集合 / 保存系 / 分岐現象

Outline of Final Research Achievements

We found new examples which exhibit wild dynamical behaviors in partially hyperbolic dynamics and homoclinic tangency. In the mechanism generating wild behavior, higher differentiabiity and information of higher differential plays important roles. In the route to find such examples, we also found a pair of Cantor sets in higher dimension which exhibits C1-stable intersection. It is contrast to the one-dimensional case where no pair of Cantor sets exhibits C1-stable intersection. We also characterized R-coveredness of a three-dimensionaltopologically transitive Anosov flow by topological information of its Birkhoff section.

Academic Significance and Societal Importance of the Research Achievements

双曲性を初めとして，力学系の位相的な性質は主にその１階微分と関係づけられることが多かったが，本研究において高階微分が力学系の周期点の数の増大度という位相的な性質が２階，３階微分と密接に関係することが明らかになった．また，０次元の集合であるカントール集合が摂動しても交わり続けると安定交差という性質は，力学系の分岐理論においてこれまでも重要な役割を果たしてきたが，本研究ではこれまで知られいたものとは全く異なるメカニズムによる安定交差の例が構成され，その分岐理論への応用がなされた．３次元アノソフ流のR-covered性の特徴づけもアノソフ流の位相的性質の理解への応用が期待される．

Research Products

• [Int'l Joint Research] Universite de Bourgogne(フランス)
• [Int'l Joint Research] the Max Plank Institute(ドイツ)
• [Int'l Joint Research] Universite de bourgogne/L'institut Fourier(フランス)
• [Int'l Joint Research] Imperial College London(英国)
• [Int'l Joint Research] Imperial College London(英国)
• [Journal Article] Stable intersection of Cantor sets in higher dimension and robust homo- clinic tangency of the largest codimension (2022)
  • Author(s): M.Asaoka
  • Journal Title: Trans. Amer. Math. Soc. Volume: 375 Issue: 02 Pages: 873-908
  • DOI: 10.1090/tran/8452
  • Peer Reviewed
• [Journal Article] Fast growth of the number of periodic points arising from heterodimensional connections (2021)
  • Author(s): Asaoka Masayuki、Shinohara Katsutoshi、Turaev Dmitry
  • Journal Title: Compositio Mathematica Volume: 157 Issue: 9 Pages: 1899-1963
  • DOI: 10.1112/s0010437x21007405
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Goodman Fried surgery, Birkhoff sections, and R-covered Anosov flows (2023)
  • Author(s): 浅岡 正幸
  • Organizer: 接触構造、特異点、微分方程式及びその周辺
  • Invited
• [Presentation] Goodman Fried surgery, Birkhoff sections, and R-covered Anosov flows (2022)
  • Author(s): 浅岡 正幸
  • Organizer: 力学系の理論と諸分野への応用
• [Presentation] Goodman Fried surgery, Birkhoff sections, and R-covered Anosov flows (2021)
  • Author(s): 浅岡 正幸
  • Organizer: 葉層構造論シンポジウム
  • Invited
• [Presentation] Stable intersection of Cantor set (2021)
  • Author(s): 浅岡 正幸
  • Organizer: 第16回代数解析幾何セミナー
  • Invited
• [Presentation] Stable intersection of Cantor sets in higher dimensions (2020)
  • Author(s): 浅岡 正幸
  • Organizer: RIMS研究集会「数理科学の諸問題と力学系理論の新展開」
• [Remarks] ArXiv: Oriented Birkhoff sections of Anosov flows
  • URL: https://arxiv.org/abs/2212.06483