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Historic behavior of wandering domains for high dimensional dynamics

Research Project

Project/Area Number 17K05283
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionTokai University

Principal Investigator

Kiriki Shin  東海大学, 理学部, 教授 (50277232)

Co-Investigator(Kenkyū-buntansha) 相馬 輝彦  東京都立大学, 理学研究科, 教授 (50154688)
中野 雄史  東海大学, 理学部, 講師 (50778313)
小川 竜  東海大学, 理学部, 講師 (90759143)
Project Period (FY) 2017-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords力学系 / 微分同相写像 / ヒストリー性 / 非双曲性 / ホモクリニック接触 / ヘテロ次元サイクル / 非双曲的集合 / 遊走領域 / ヒストリック振る舞い / historic behavior / homoclinic class / diffeomorphism / heterodimensional cycle / 微分同相写像系 / ホモクリニック クラス / ホモクリニック分岐 / Bogdanov-Takens分岐 / トポロジー / 解析学
Outline of Final Research Achievements

Results were obtained that largely satisfied the original research objectives. Specifically, we have succeeded to show that C1-generically for diffeomorphisms of manifolds of dimension d at least 3, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behaviour. The results were published in one of the leading journals of the American Mathematical Society.

Academic Significance and Societal Importance of the Research Achievements

本研究は『Takensの最終問題』として知られる公開問題を解くために多くの情報を与えている.『Takensの最終問題』とは「ヒストリックな振舞いを持つ軌道の初期点集合がルベーグ測度正であるような微分同相写像が持続的に存在するか?」というものである.これに対して我々は2017年にNewhouse領域に属する任意の2次元微分同相写像は,Takensの最終問題に対し肯定的な性質をもつ微分同相写像でCr近傍(ただしrは2以上かつ無限大未満)できることをしめした.これは2次元に限った結果であったが,本研究の成果は3次元以上のヒストリックな振舞いの存在を示したという点で力学系的にとても重要な結果である.

Report

(5 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (13 results)

All 2020 2019 2018 2017 Other

All Int'l Joint Research (2 results) Journal Article (5 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 5 results,  Acknowledgement Compliant: 1 results) Presentation (5 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results) Book (1 results)

  • [Int'l Joint Research] University Federal Fluminense-UFF(ブラジル)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University Federal Fluminense(ブラジル)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Historic behavior in nonhyperbolic homoclinic classes2019

    • Author(s)
      Barrientos Pablo G.、Kiriki Shin、Nakano Yushi、Raibekas Artem、Soma Teruhiko
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 148 Issue: 3 Pages: 1195-1206

    • DOI

      10.1090/proc/14809

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Historic behaviour for nonautonomous contraction mappings2019

    • Author(s)
      S. Kiriki, Y. Nakano and T. Soma
    • Journal Title

      Nonlinearity

      Volume: 32 Issue: 3 Pages: 1111-1124

    • DOI

      10.1088/1361-6544/aaf253

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Moduli of 3-dimensional diffeomorphisms with saddle-foci2018

    • Author(s)
      Shinobu Hashimoto, Shin Kiriki, Teruhiko Soma
    • Journal Title

      Discrete & Continuous Dynamical Systems - A

      Volume: 38 Issue: 10 Pages: 5021-5037

    • DOI

      10.3934/dcds.2018220

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Non-trivial wandering domains for heterodimensional cycles2017

    • Author(s)
      Shin Kiriki、Yushi Nakano 、Teruhiko Soma
    • Journal Title

      Nonlinearity

      Volume: 30 Issue: 8 Pages: 3255-3270

    • DOI

      10.1088/1361-6544/aa7cc6

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Takens' last problem and existence of non-trivial wandering domains2017

    • Author(s)
      Kiriki, Shin; Soma, Teruhiko
    • Journal Title

      Advances in Mathematics

      Volume: 306 Pages: 524-588

    • DOI

      10.1016/j.aim.2016.10.019

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Historic behavior in non-hyperbolic homoclinic classes2020

    • Author(s)
      桐木 紳
    • Organizer
      2019年度 冬の力学系研究集会
    • Related Report
      2019 Research-status Report
  • [Presentation] Historic behavior for heterodimensional cycles2019

    • Author(s)
      桐木 紳
    • Organizer
      冬の力学系研究集会
    • Related Report
      2018 Research-status Report
  • [Presentation] Robust historic behavior of generic orbits for heterodimensional cycles2018

    • Author(s)
      Shin KIRIKI
    • Organizer
      One day Workshop of Dynamical Systems in Suzhou
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] C^2-robust heterodimensional tangencies of codimension 22018

    • Author(s)
      Shin Kiriki
    • Organizer
      2017年度 冬の力学系研究集会
    • Related Report
      2017 Research-status Report
  • [Presentation] Historic Behavior for Non-trivial Wandering Domains2017

    • Author(s)
      Shin Kiriki
    • Organizer
      2017 NCTS Workshop on Dynamical Systems
    • Related Report
      2017 Research-status Report
  • [Book] 数学の問題をうまくきれいに解く秘訣2017

    • Author(s)
      桐木 紳, 桐木 由美
    • Total Pages
      208
    • Publisher
      共立出版
    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2022-01-27  

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