A study on the dynamics of two dimensional polynomial skew products

• Project/Area Number: 21540203
• 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 東京工芸大学, 工学部, 教授 (50172359)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 複素解析 / 複素力学系 / Axiom A / skew product / base Julia集合 / fiber Julia集合 / saddle basic set / relation / holomorphic motion / polynomial skew product / base Julia set / solenoid / 危点集合 / 安定多様体 / 不安定多様体

Research Abstract

Investigating the link between postcritical behaviors and the relations of saddle basic sets for Axiom A polynomial skew products on C2, we characterized some properties of the three kinds of accumulation sets of the critical set in terms of the saddle basic sets. We also showed that these properties are preserved in the hyperbolic components. We gave an example with a property characteristic to the higher degree maps.

Research Products

• [Journal Article] Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C2 (2013)
  • Author(s): 中根静男
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 33 Issue: 4 Pages: 1124-1145
  • DOI: 10.1017/s0143385712000193
  • Peer Reviewed
• [Journal Article] A novel method for evaluating postsurgical results of unilateral cleft lip and palate with theuse of Hausdorff distance. (2012)
  • Author(s): Toru Yanagawa, Shizuo Nakane, et.al
  • Journal Title: Oral Surg OralMed Oral Pathol Oral Radiol Volume: Vol.114 Issue: 6 Pages: 704-711
  • DOI: 10.1016/j.oooo.2012.01.042
  • Peer Reviewed
• [Journal Article] Presurgical nasoalveolar molding orthopedic treatment improves the outcome of primary cheiloplasty of unilateral complete cleft lip and palate, as assessed by naris morphology and cleft gap. (2012)
  • Author(s): Toru Yanagawa, Shizuo Nakane, et.al.
  • Journal Title: Journal of Craniofacial Surgery Volume: Vol. 23 Issue: 6 Pages: 1596-1601
  • DOI: 10.1097/scs.0b013e31825196dc
  • Peer Reviewed
• [Journal Article] On saddle basic sets forAxiom A polynomial skew products on C^2. (2012)
  • Author(s): Shizuo Nakane
  • Journal Title: 数理解析研究所講究録 Volume: Vol. 1807 Pages: 66-73
• [Journal Article] On saddle basic sets for Axiom A polynomial skew products on C^2 (2012)
  • Author(s): Shizuo Nakane
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1807 Pages: 66-73
• [Journal Article] Presurgical nasoalveolar molding orthopedic treatment improves the outcome of primary cheiloplasty of unilateral complete cleft lip and palate, as assessed by naris morphology and cleft gap (2012)
  • Author(s): Hiroyoshi Sasaki
  • Journal Title: Journal of Craniofacial Surgery Volume: 23 Pages: 1596-1601
  • Peer Reviewed
• [Journal Article] Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C^2. (2011)
  • Author(s): Shizuo Nakane
  • Journal Title: 数理解析研究所講究録 Volume: Vol. 1762 Pages: 114-119
• [Journal Article] Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C^2 (2011)
  • Author(s): Shizuo Nakane
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1762 Pages: 114-119
• [Journal Article] Component-wise accumu- lation sets for Axiom A polynomial skew products. (2010)
  • Author(s): Shizuo Nakane
  • Journal Title: 数理解析研究所講究録 Volume: Vol. 1699 Pages: 81-87
• [Journal Article] Component-wise accumulation sets for Axiom A polynomial skew products (2010)
  • Author(s): Shizuo Nakane
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1699 Pages: 81-87
• [Journal Article] Remarks on Axiom A poly- nomial skew products of higher degrees. (2009)
  • Author(s): Shizuo Nakane
  • Journal Title: Proceedings of ICFIDCAA Volume: Vol.16 Pages: 196-200
  • Peer Reviewed
• [Journal Article] Branner-Hubbard- Lavaurs deformations for real cubic poly- nomials with a parabolic fixed point. (2009)
  • Author(s): Shizuo Nakane
  • Journal Title: Conformal Geometry and Dynamics Volume: Vol. 13 Issue: 5 Pages: 110-123
  • DOI: 10.1090/s1088-4173-09-00192-1
  • Peer Reviewed
• [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 13 Pages: 110-123
  • Peer Reviewed
• [Journal Article] Remarks on Axiom A polynomial skew products of higher degrees (2009)
  • Author(s): Shizuo Nakane
  • Journal Title: Proceedings of 16^<th> ICFIDCAA Pages: 196-200
• [Presentation] On saddle basic sets for Axiom A polynomial skew products on C2. RIMS 研究集会 (2012)
  • Author(s): Shizuo Nakane
  • Organizer: 複素力学系の総合的研究
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-01-26
• [Presentation] On saddle basic sets for Axiom A polynomial skew products on C^2 (2012)
  • Author(s): Shizuo Nakane
  • Organizer: 研究集会「複素力学系の総合的研究」
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2012-01-26
• [Presentation] Postcritical sets and saddle basic sets for Axiom A polynomialskew products on C2. RIMS 研究集会 (2010)
  • Author(s): Shizuo Nakane
  • Organizer: 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: 2010-12-09
• [Presentation] Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C^2 (2010)
  • Author(s): Shizuo Nakane
  • Organizer: 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
  • Year and Date: 2010-12-09
• [Presentation] Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C2. (2010)
  • Author(s): Shizuo Nakane
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-23
• [Presentation] Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C^2 (2010)
  • Author(s): Shizuo Nakane
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-23
• [Presentation] Dynamics of Axiom Apolynomial skew products (2010)
  • Author(s): Shizuo Nakane
  • Organizer: 国際数学者会議
  • Place of Presentation: インド, ハイデラバード
  • Year and Date: 2010-08-24
• [Presentation] Dynamics of Axiom A polynomial skew products (2010)
  • Author(s): Shizuo Nakane
  • Organizer: 国際数学者会議2010
  • Place of Presentation: ハイデラバード
  • Year and Date: 2010-08-24
• [Presentation] 複素力学系とその 関連分野の総合的研究 RIMS 研究集会 (2009)
  • Author(s): Shizuo Nakane
  • Organizer: Component-wise accumu- lation sets for Axiom A polynomial skew products.
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-12-16
• [Presentation] Component-wise accumulation sets of critical sets for Axiom A polynomial skew products (2009)
  • Author(s): 中根静男
  • Organizer: 研究集会「複素力学系とその関連分野の総合的研究」
  • Place of Presentation: 京都大学
  • Year and Date: 2009-12-16
• [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