A study on the dynamics of the family of complex cubic polynomials

• Project/Area Number: 17540177
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokyo Polytechnic University
• Principal Investigator: NAKANE Shizuo Tokyo Polytechnic University, 工学部, 教授 (50172359)
• Project Period (FY): 2005 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,060,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥360,000); Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000); Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: Branner-Hubbard-Lavaurs deformation / stretching ray / straightening map / Axiom A polynomial skew product / base Julia set / fiber Julia set / point-wise accumulation set / component-wise accumulation set / parabolic locus / stretching map / Axiom A / polynomial skew product / 集積点集合 / pointwise accumulation set / Chebyshev写像 / 実biquadratic polynomials / bifurcation current / Branner-Hubbard deformation / central hyperbolic component / monodromy / Boettcher-Lavaurs vector / bifurcation measure

Research Abstract

放物的不動点を持つ3次多項式のBranner-Hubbard-Lavaurs deformationの理論を構築し、stretching raysの振動の規則性やstretching mapの不連続性を解明した。高次多項式のJulia setの性質を用いることにより、base Julia set が連結でないAxiom Apolynomial skew productの力学系について、特に危点集合の集積点集合の性質を解明し、新しい例を構成した。

Research Products

• [Journal Article] Remarks on Axiom A polynomial skew products of higher degrees. Proc (2009)
  • Author(s): Shizuo Nakane
  • Journal Title: International Conference on Finite or Infinite Dimensional Complex Analysis and Applications Pages: 196-200
• [Journal Article] Branner-Hubbard-Lavaurs deformations for real cubic polynomials with a parabolic fixed point (2009)
  • Author(s): Shizuo Nakane
  • Journal Title: Conformal Geometry and Dynamics Vol.13 Pages: 110-123
  • Peer Reviewed
• [Journal Article] External rays for polynomial maps of two variables associated with Chebyshev maps (2008)
  • Author(s): Shizuo Nakane
  • Journal Title: Jounal of Mathematical Analysis and Applications Vol.338 Pages: 552-562
  • Peer Reviewed
• [Journal Article] Branner-Hubbard-Lavaurs deformation of parabolic cubic polynomials (2008)
  • Author(s): Shizuo Nakane
  • Journal Title: Proc. 15^<th> Int. Conf. on Fin. or Inf. Dim. Complex Anal. and Appl. Pages: 301-306
• [Journal Article] External rays for polynomial maps of two variables associated with Chebyshev maps (2008)
  • Author(s): Shizuo Nakane
  • Journal Title: Journal of Mathematical Analysis and Applications 338 Pages: 552-562
  • Peer Reviewed
• [Journal Article] A mathematical study of Carnot cycles for real gasses (2007)
  • Author(s): Shizuo Nakane
  • Journal Title: JP Jour. Solids & Structures Vol.1 Pages: 27-33
  • Peer Reviewed
• [Journal Article] A mathematical study of Carnot cycles for real gases (2007)
  • Author(s): Shizuo Nakane
  • Journal Title: JP Journal of Solids and Structures 1 Pages: 27-33
  • Peer Reviewed
• [Journal Article] Branner-Hubbard-Lavaurs deformation of parabolic cubic polynomials (2006)
  • Author(s): Shizuo Nakane
  • Journal Title: 京都大学数理解析研究所講究録 1494 Pages: 96-98
• [Journal Article] Non-landing of stretching rays for real cubic polynomials and real biquadratic polynomials (2006)
  • Author(s): Shizuo Nakane
  • Journal Title: 数理解析研究所講究録 1447 Pages: 164-169
• [Journal Article] Dynamics of a family of quadratic maps in the quaternion space (2005)
  • Author(s): Shizuo Nakane
  • Journal Title: International Journal of Bifurcation and Chaos Vol.15 Pages: 2535-2543
  • NAID: 110000272991
  • Peer Reviewed
• [Journal Article] Dynamics of a family of quadratic maps in the quaternion space (2005)
  • Author(s): Shizuo Nakane
  • Journal Title: International Journal of Bifurcation and Chaos 15 Pages: 2535-2543
  • NAID: 110000272991
• [Presentation] On Axiom A polynomial skew products of higher degrees (2008)
  • Author(s): Shizuo Nakane
  • Organizer: Workshop "Moduli and invariants in complex analysis and algebraic geometry."
  • Place of Presentation: 京都工芸繊維大学
  • Year and Date: 2008-11-27
• [Presentation] Axiom A polynomial skew products of higher degrees (2008)
  • Author(s): Shizuo Nakane
  • Organizer: RIMS Workshop"New development in the study of Dynamical Systems beyond Uniform-Hyperbolicity."
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-10-02
• [Presentation] Axiom A polynomial skew products of higher degrees (2008)
  • Author(s): Shizuo Nakane
  • Organizer: Workshop“New Development in the Study of Dynnam. Sys. Beyond Hyperbolicity"
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-10-02
• [Presentation] Component-wise accumulation sets of critical sets for Axiom A polynomial skew products (2008)
  • Author(s): 中根静男
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-25
• [Presentation] Remarks on Axiom A polynomial skew products of higher degrees (2008)
  • Author(s): Shizuo Nakane
  • Organizer: 16^<th> International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: 韓国, 慶州, 東国大学校
  • Year and Date: 2008-07-29
• [Presentation] Remarks on Axiom A polynomial skew products of higher degrees (2008)
  • Author(s): Shizuo Nakane
  • Organizer: 16^<th> International Conference on Fin. or Inf. Dim. Complex Anal. and Appl.
  • Place of Presentation: 東国大学校(韓国)
  • Year and Date: 2008-07-29
• [Presentation] An Axiom A polynomial skew product of degree four (2008)
  • Author(s): 中根静男
  • Organizer: 日本数学会年会
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2008-03-24
• [Presentation] An Axiom A polynomial skew product of degree four (2008)
  • Author(s): Shizuo Nakane
  • Organizer: 日本数学会年会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-24
• [Presentation] Branner-Hubbard-Lavaurs deformation of parabolic cubic polynomials (2007)
  • Author(s): Shizuo Nakane
  • Organizer: 15^<th> International Congress on Finite or Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2007-07-31
• [Presentation] Branner-Hubbard-Lavaurs deformation of parabolic cubic polynomials (2007)
  • Author(s): Shizuo Nakane
  • Organizer: 15th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2007-07-31
• [Presentation] Branner-Hubbard-Lavaurs deformation of parabolic cubic polynokmials (2006)
  • Author(s): 中根静男
  • Organizer: RIMS研究集会"複素力学系とその周辺分野の研究"
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2006-01-25
• [Presentation] Branner-Hubbard-Lavaurs motion of parabolic cubic polynomials (2005)
  • Author(s): Shizuo Nakane
  • Organizer: Workshop"Parabolic Points"in Hangzhou
  • Place of Presentation: 中国, 杭州, 浙江大学
  • Year and Date: 2005-04-18
• [Remarks]
  • URL: http://www.gen.t-kougei.ac.jp/math/nakane/index.html
• [Remarks]
  • URL: http://www.gen.t-kougei.ac.jp/math/nakane.index.html