Dynamical studyof the parameter space for the complexHenon family

2012 Fiscal Year Final Research Report

• Project/Area Number: 21740125
• 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 九州大学, 数理学研究院, 准教授 (20304727)
• Project Period (FY): 2009 – 2012
• Keywords: 複素力学系 / ジュリア集合 / iterated monodromy group

Research Abstract

Although the complex Henon map is regarded as one of the most standard holomorphic dynamical systems in complex dimension two, the dynamical studyof its Julia set from the topological and combinatorial viewpoint has not been well developed. We introduced a combinatorial model called the Hubbard tree for such map and developed an algebraic formalism called the iterated monodromy group. Moreover, together with the help of numerical experiments, we also analyzed the dynamical structurein the parameter space of the Henon family.

Research Products

• [Journal Article] Corrigendum, Adv (2011)
  • Author(s): Yutaka Ishii
  • Journal Title: Math. 226 Volume: no. 4 Pages: 3850-3855
• [Journal Article] Homotopy shadowing. (2010)
  • Author(s): Yutaka Ishiiand John Smillie
  • Journal Title: Amer. J. Math Volume: no. 4 Pages: 987-1029
• [Journal Article] Hyperbolic polynomial diffeomorphisms of C^2. II: Hubbard trees (2009)
  • Author(s): Yutaka Ishii
  • Journal Title: Adv. Math. 220 Volume: no. 4 Pages: 985-1022
• [Journal Article] Julia sets and chaotic tunneling I (2009)
  • Author(s): Akira Shudo, Yutaka Ishii, Kensuke Ikeda
  • Journal Title: J. Phys. A: Math. Theor Volume: 42 Pages: 26
• [Journal Article] Julia sets and chaotic tunneling II (2009)
  • Author(s): Akira Shudo, Yutaka Ishii, Kensuke Ikeda
  • Journal Title: J. Phys. A: Math. Theor Volume: 42 Pages: 34
• [Presentation] Homotopy kernel (2012)
  • Author(s): Yutaka Ishii
  • Organizer: New Developments in Complex Dynamical Systems
  • Place of Presentation: RIMS at Kyoto University, Kyoto, Japan
  • Year and Date: 20121010-14
• [Presentation] Trinitas for complex Henon maps (2012)
  • Author(s): Yutaka Ishii
  • Organizer: Workshop on Holomorphic Vector Fields and Foliations
  • Place of Presentation: Ryukoku University Seminar House, Kyoto, Japan
  • Year and Date: 20121007-09
• [Presentation] Iterated monodromy groups for complex Henon maps (2011)
  • Author(s): Yutaka Ishii
  • Organizer: Automorphisms of Algebraic Varieties
  • Place of Presentation: Hotel Laforet, Shirahama, Japan
  • Year and Date: 20111019-23
• [Presentation] Hubbard trees for complex Henon maps (2011)
  • Author(s): Yutaka Ishii
  • Organizer: Symposium on Complex Analysis
  • Place of Presentation: Chiba University, Chiba, Japan
  • Year and Date: 20111008-10
• [Presentation] Iterated monodromy groups for complex Henon maps (2011)
  • Author(s): Yutaka Ishii
  • Organizer: Gottingen-Penn State International Summer School
  • Place of Presentation: ottingen, Germany
  • Year and Date: 20110807-14
• [Presentation] Some conjectures on the parameter space of the Lozi family (2011)
  • Author(s): Yutaka Ishii
  • Organizer: Kyoto Symposium on Complex Analysis in Several Variables XIV
  • Place of Presentation: Kyoto University, Kyoto, Japan
  • Year and Date: 20110718-21
• [Presentation] Some conjectures on the parameter space of the Lozi family (2011)
  • Author(s): Yutaka Ishii
  • Organizer: SIAM Conference on Applications of Dynamical Systems
  • Place of Presentation: Snowbird Ski and Summer Resort, Snowbird, USA
  • Year and Date: 20110521-26
• [Remarks]
  • URL: https://sites.google.com/site/webpageyutakaishii/