Dynamics of complex Henon maps: from dynamical space to parameter space

• Project/Area Number: 25287020
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Ishii Yutaka 九州大学, 数理学研究院, 准教授 (20304727)
• Co-Investigator(Kenkyū-buntansha): 浅岡 正幸 京都大学, 理学研究科, 准教授 (10314832) ; 荒井 迅 中部大学, 創発学術院, 教授 (80362432)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000); Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2013: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
• Keywords: ヘノン写像 / ジュリア集合 / ホースシュー / 反復写像系 / Henon 写像 / 精度保証計算 / マンデルブロー集合 / 複素ヘノン写像 / モノドロミー作用 / 双曲的ホースシュー / Hubbard tree / エノン写像 / ホースシュー領域 / 複素力学系 / 変分原理 / Julia 集合 / homotopy kernel / ハバード木 / 反復モノドロミー群

Outline of Final Research Achievements

We obtained some results concerning combinatorial and topological aspects of both real and complex Henon maps. Among others, we studied topological properties of the hyperbolic horseshoe locus and the maximal entropy locus in the parameter space of the real Henon family, variational characterization of some invariant measures of the Henon maps at the boundary of the horseshoe locus, and the representation of the monodromy group action of the horseshoe locus for the complex Henon family to the automorphism group of the corresponding symbolic dynamics.

Research Products

• [Journal Article] M_4 is regular-closed (2018)
  • Author(s): Yutaro Himeki and Yutaka Ishii
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 印刷中 Issue: 1 Pages: 213-220
  • DOI: 10.1017/etds.2018.27
  • Peer Reviewed
• [Journal Article] On parameter loci of the Henon family (2018)
  • Author(s): Zin Arai and Yutaka Ishii
  • Journal Title: Communications in Mathematical Physics Volume: 印刷中
  • NAID: 120006940730
  • Peer Reviewed
• [Journal Article] Dynamics of polynomial diffeomorphisms of C^2: Combinatorial and topological aspects (2017)
  • Author(s): Yutaka Ishii
  • Journal Title: Arnold Mathematical Journal Volume: 印刷中
  • NAID: 120006940731
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces (2016)
  • Author(s): Masayuki Asaoka, Kei Irie
  • Journal Title: Geometric and Functional Analysis Volume: 26 Issue: 5 Pages: 1245-1254
  • DOI: 10.1007/s00039-016-0386-3
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics (2016)
  • Author(s): M. Asaoka, K. Shinohara, D.Turaev
  • Journal Title: Mathematische Annalen Volume: - Issue: 3-4 Pages: 1277-1309
  • DOI: 10.1007/s00208-016-1468-0
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Hyperbolic polynomial diffeomorphisms of C^2. III: Iterated monodromy groups (2014)
  • Author(s): Yutaka Ishii
  • Journal Title: Advances in Mathematics Volume: 255 Pages: 242-304
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] On some conjectures of Chris Lipa (2017)
  • Author(s): 石井豊
  • Organizer: 多変数の複素力学系と函数論
  • Place of Presentation: 京都大学
  • Int'l Joint Research / Invited
• [Presentation] ヘノン写像族のホースシュー領域について (2016)
  • Author(s): 石井豊
  • Organizer: エルゴード理論研究集会
  • Place of Presentation: 筑波大学
  • Invited
• [Presentation] Tight automata for some quadratic polynomials (2015)
  • Author(s): Yutaka Ishii
  • Organizer: Workshop on Symbolic Dynamics and Related Topics
  • Place of Presentation: Hiroshima, Japan
  • Year and Date: 2015-10-23
  • Invited
• [Presentation] On parameter loci of the Henon family (2015)
  • Author(s): Yutaka Ishii
  • Organizer: School and Conference on Dynamical Systems
  • Place of Presentation: Trieste, Italy
  • Year and Date: 2015-08-01
  • Invited
• [Presentation] On parameter loci of the Henon family (2015)
  • Author(s): Yutaka Ishii
  • Organizer: Geometric Topology, Geometric Group Theory & Dynamical Systems
  • Place of Presentation: Dubrovnik, Croatia
  • Year and Date: 2015-06-23
  • Invited