Complex dynamics on K3 surfaces as conservative systems

• Project/Area Number: 18740091
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Kyushu University
• Principal Investigator: ISHII Yutaka Kyushu University, 数理学研究院, 准教授 (20304727)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥2,940,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥800,000 (Direct Cost: ¥800,000); Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: K3曲面 / 複素力学系 / ジュリア集合 / KAM不変円 / 体積保存系 / 自己相似性 / エノン写像 / エントロピー

Research Abstract

K3曲面上の正則自己同型がなす複素力学系およびそれに関連した課題について、体積保存力学系の立場から研究を行った。まずK3曲面上の複素力学系については、Mazurの(2,2,2)-typeのK3曲面の実部であってその上の力学系が可積分に見える(2点をのぞいて不変な円周で曲面が埋め尽くされる)を発見していたが、この例の場合、代数的な第一積分は存在しないことをSerge Cantat 氏とともに証明した。また、特に体積保存系特有の現象としてKAM circleの崩壊の順序構造や自己相似性の問題が古くから物理学者によって取り上げられていたが、それらの現象と複素力学系の不変集合であるジュリア集合との関係を論じ、いくつかの予想をたてた。

Research Products

• [Journal Article] Hyperbolic polynomial diffeomorphisms of C^2. II : Hubbard trees (2009)
  • Author(s): YUTAKA ISHII
  • Journal Title: Advances in Mathematics Vol 220 Pages: 985-1022
  • Peer Reviewed
• [Journal Article] Homotopy shadowing (2009)
  • Author(s): YUTAKA ISHII and John Smillie
  • Journal Title: To appear in Amer. J. Math
• [Journal Article] Hyperbolic polguowial diffeoworphisms of C^2 II : Habbard trees (2009)
  • Author(s): Yutaka Ishii
  • Journal Title: Advauces iu Mathewatics 220 Pages: 985-1022
  • Peer Reviewed
• [Journal Article] Hyperbolic polynomial difeomorphisms of C^2. I : A non-planar map (2008)
  • Author(s): YUTAKA ISHII
  • Journal Title: Advances in Mathematics Vol 218 Pages: 417-464
  • Peer Reviewed
• [Journal Article] Hyperbolic polguowial diffeoworphisms of C^2 I : Anon-planarmap (2008)
  • Author(s): Yutaka Ishii
  • Journal Title: Advauces iu Mathewatics 218 Pages: 417-464
  • Peer Reviewed
• [Journal Article] Hyperbolic pdynowuall diffeomphismsof C^2: I. (2008)
  • Author(s): Yutaka, Ishii
  • Journal Title: Advanw in Mathematics 218 Pages: 417-464
  • Peer Reviewed
• [Journal Article] Lap number entropy formula for piecewise affine and projective maps in several dimensions (2007)
  • Author(s): YUTAKA ISHII and Duncan Sands
  • Journal Title: Nonlinearity Vol 20 Pages: 2755-2772
  • Peer Reviewed
• [Journal Article] Lap number eutropy frmala for piecewise affine meps inseveral dima sious. (2007)
  • Author(s): Yutaka, Ishii・duucan sands
  • Journal Title: Noulinearity 20 Pages: 2755-2772
  • Peer Reviewed
• [Presentation] Dynamics of hyperbolic complex Henon maps (2007)
  • Author(s): YUTAKA ISHII
  • Organizer: Seminar on complex analysis in several complex variable
  • Place of Presentation: Kumamoto University, Japan
• [Presentation] Hubbard trees for complex Henon maps (2007)
  • Author(s): YUTAKA ISHII
  • Organizer: Workshop on dynamical systems and foliations
  • Place of Presentation: University of Tokyo, Japan
• [Presentation] Hubbard trees for complex Henon maps (2007)
  • Author(s): YUTAKA ISHII
  • Organizer: Seminar on Teichmuller spaces
  • Place of Presentation: Universite de Provence, France
• [Presentation] Hubbard trees for complex Henon maps (2007)
  • Author(s): YUTAKA ISHII
  • Organizer: Birational geometry and complex dynamical systems
  • Place of Presentation: Nagoya University, Japan
• [Presentation] Hubbard trees for complex Henon maps (2006)
  • Author(s): YUTAKA ISHII
  • Organizer: Workshop dedicated for the 60th birthday of John Hubbard
  • Place of Presentation: CIRM at Luminy, France