Study of diffeomorphisms on geometric 3-manifolds

• Project/Area Number: 26400093
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: Soma Teruhiko 首都大学東京, 理学研究科, 教授 (50154688)
• Co-Investigator(Kenkyū-buntansha): 今井 淳 千葉大学, 大学院理学研究院, 教授 (70221132)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 3-manifold / dynamical systems / diffeomorphism / 微分同相写像 / 微分同相群 / 3次元多様体 / 遊走領域 / 双曲多様体 / クライン群 / 歴史挙動 / 歴史的挙動

Outline of Final Research Achievements

The main aim of this research project is to study structures of diffeomorphisms on 3-dimensional manifolds. To accomplish it, we investigated 2 and 3-dimensional diffeomorphisms from dynamical point of view. In fact we have obtained some new results on the existence of wandering domains including　the affirmative answer of F. Takens conjecture. This is a research concerning on the complexity of orbits. This shows theoretically that it is often impossible to predict the orbits from statistical point of view.

Academic Significance and Societal Importance of the Research Achievements

この研究は，純粋に数学的な見地からの研究であるが，ある種の運動している物体の軌道を推測することが難しいことを理論的に説明してるとも言える．

Research Products

• [Journal Article] Historic behaviour for nonautonomous contraction mappings (2019)
  • 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
  • Peer Reviewed
• [Journal Article] Moduli of 3-dimensional diffeomorphisms with saddle-foci (2018)
  • 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
  • Peer Reviewed
• [Journal Article] Non-trivial wandering domains for heterodimensional cycles (2017)
  • Author(s): Shin Kiriki、Yushi Nakano 、Teruhiko Soma
  • Journal Title: Nonlinearity Volume: 30 Issue: 8 Pages: 3255-3270
  • DOI: 10.1088/1361-6544/aa7cc6
  • Peer Reviewed
• [Journal Article] Takens' last problem and existence of non-trivial wandering domains (2017)
  • Author(s): Kiriki, Shin; Soma, Teruhiko
  • Journal Title: Advances in Mathematics Volume: 306 Pages: 524-588
  • DOI: 10.1016/j.aim.2016.10.019
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Geometric Lorenz flows with historic behavior (2016)
  • Author(s): Kiriki, Shin; Li, Ming-Chia; Soma, Teruhiko
  • Journal Title: Discrete and Continuous Dynamical Systems. Series A Volume: 36 Issue: 12 Pages: 7021-7028
  • DOI: 10.3934/dcds.2016105
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Möbius invariant energies and average linking with circles (2015)
  • Author(s): Jun O'Hara（今井淳）, Gil Solanes
  • Journal Title: Tohoku Math. Journal Volume: 67 Issue: 1 Pages: 51-82
  • DOI: 10.2748/tmj/1429549579
  • NAID: 130007961973
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Minimal unfolded regions of a convex hull and parallel bodies (2015)
  • Author(s): Jun O'Hara（今井淳）
  • Journal Title: Hokkaido Math. Journal Volume: 44 Pages: 175-183
  • Peer Reviewed
• [Journal Article] Möbius invariant energies and average linking with circles (2015)
  • Author(s): Jun O'Hara（今井淳）, Gil Solanes
  • Journal Title: Tohoku Math. Journal Volume: 67 Pages: 51-82
  • Peer Reviewed
• [Presentation] Volume and structure of hyperbolic 3-manifolds (2017)
  • Author(s): Teruhiko Soma
  • Organizer: JNU-KAIST Geometric Topology Fair
  • Invited
• [Presentation] 既約3次元多様体の幾何とトポロジー (2016)
  • Author(s): 相馬輝彦
  • Organizer: 日本数学会秋期総合分科会
  • Place of Presentation: 関西大学
  • Year and Date: 2016-09-16
  • Invited
• [Presentation] Geometric Lorenz flows with historic behavior (2016)
  • Author(s): 相馬輝彦
  • Organizer: 2015年度冬の力学系研究集会
  • Place of Presentation: 日本大学軽井沢研修所（長野県，軽井沢町）
  • Year and Date: 2016-01-10
• [Presentation] Regularization of energies of knots and surfaces (2015)
  • Author(s): Jun O'Hara（今井淳）
  • Organizer: Geometric energies with links to applications, topology and open problems
  • Place of Presentation: University of Basel (Basel, Switzerland)
  • Year and Date: 2015-09-03
  • Invited
• [Presentation] Energy of knots, regularization and analytic continuations (2015)
  • Author(s): Jun O'Hara（今井淳）
  • Organizer: Seminaris Geometria
  • Place of Presentation: バルセロナ（スペイン）
  • Year and Date: 2015-03-17
  • Invited
• [Book] 数学セミナー(2016-5) (2016)
  • Author(s): 相馬輝彦
  • Total Pages: 5
  • Publisher: 日本評論社
• [Remarks] Soma-index
  • URL: http://www.comp.tmu.ac.jp/trhksoma/
• [Remarks] 教員紹介 相馬 輝彦 | 首都大学東京
  • URL: https://www.tmu.ac.jp/stafflist/data/sa/570.html
• [Remarks] 教員紹介 相馬輝彦
  • URL: https://www.tmu.ac.jp/stafflist/data/sa/570.html
• [Remarks] 論文 （1997年以降： 国内 海外）
  • URL: http://www.comp.tmu.ac.jp/trhksoma/papers.html