Thermodynamic formalism for the Henon map at the first bifurcation

• Project/Area Number: 23740121
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: Keio University (2013-2014); Kyoto University (2012); The University of Tokyo (2011)
• Principal Investigator: TAKAHASI Hiroki 慶應義塾大学, 理工学部, 講師 (00467440)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 力学系 / エノン写像 / 国際研究者交流 / 韓国 / 双曲性 / 分岐 / エルゴード理論 / 国際情報交換 / アメリカ / 熱力学形式

Outline of Final Research Achievements

We investigated the dynamics of the Henon map at the first bifurcation parameter with the use of themodynamic fomalism, and proved the existence and uniqueness of equilibrium measures. We also derived a version of Bowen-Manning formula which connects the Hausdorff dimension of the invariant set with the entropy and Lyapunov exponent of invariant measures. We also proved nice statistical properties of equilibrium measures, such as an exponential decay of correlations and the central limit theorem.

Research Products

• [Journal Article] Equilibrium measures for the Henon map at the first bifurcation: uniqueness and geometric/statistical properties (2015)
  • Author(s): Samuel Senti, Hiroki Takahasi
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: to appear
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Equilibrium measures for the Henon map at the first bifurcation: uniqueness and geometric/statistical properties (2014)
  • Author(s): Samuel Senti and Hiroki Takahasi
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 掲載予定
  • Peer Reviewed
• [Journal Article] Equilibrium measures for the Henon map at the first bifurcation (2013)
  • Author(s): Samuel Senti and Hiroki Takahasi
  • Journal Title: Nonlinearity Volume: 26 Issue: 6 Pages: 1719-1741
  • DOI: 10.1088/0951-7715/26/6/1719
  • Peer Reviewed
• [Journal Article] Multifractal formalism for Benedicks-Carleson quadratic maps (2013)
  • Author(s): Yong Moo Chung and Hiroki Takahasi
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: published online Issue: 4 Pages: 1-26
  • DOI: 10.1017/etds.2012.188
  • Peer Reviewed
• [Journal Article] Prevalent Dynamics at the First Bifurcation of Henon-like Families (2012)
  • Author(s): Hiroki Takahasi
  • Journal Title: Communications in Mathematical Physics Volume: 312 Issue: 1 Pages: 37-85
  • DOI: 10.1007/s00220-012-1442-y
  • Peer Reviewed
• [Journal Article] Statistical Properties of Nonuniformly Expanding 1d Maps with Logarithmic Singularities (2012)
  • Author(s): Hiroki Takahasi
  • Journal Title: Noninearity Volume: 25 Issue: 2 Pages: 551-567
  • DOI: 10.1088/0951-7715/25/2/551
  • Peer Reviewed
• [Journal Article] Nonuniformly Expanding 1d Maps with Logarithmic Singularities (2012)
  • Author(s): Hiroki Takahasi and Qiudong Wang
  • Journal Title: Nonlinearity Volume: 25 Issue: 2 Pages: 533-550
  • DOI: 10.1088/0951-7715/25/2/533
  • Peer Reviewed
• [Journal Article] Large Deviation Principle for Benedicks-Carleson Quadratic Maps (2012)
  • Author(s): Hiroki Takahasi and Yong Moo Chung
  • Journal Title: Communications in Mathematical Physics Volume: in press
  • Peer Reviewed
• [Presentation] Lyapunov spectrum of Henon-like maps at the first bifurcation (2014)
  • Author(s): Hiroki Takahasi
  • Organizer: ICM Satellite Conference ``Dynamical Systems and Related Topics"
  • Place of Presentation: Chungnam大学（韓国、ソウル市）
  • Year and Date: 2014-08-08
  • Invited
• [Presentation] Large deviation principle for Benedicks-Carleson quadratic maps (2012)
  • Author(s): 高橋博樹
  • Organizer: Workshop on Non-uniformly Hyperbolic and Neutral One-dimensional Dynamics
  • Place of Presentation: National University of Singapore
  • Invited
• [Presentation] Multifractal analysis for the Henon map at the first bifurcation (2012)
  • Author(s): 高橋博樹
  • Organizer: Ergodic Theory and Metric Number Theory
  • Place of Presentation: 日本女子大学
• [Presentation] Multifractal Formalism for Benedicks-Carleson Quadratic Maps (2012)
  • Author(s): Hiroki Takahasi
  • Organizer: Dynamics of Complex Systems(招待講演)
  • Place of Presentation: 北海道大学
• [Presentation] 統計力学的手法による馬蹄の大域分岐問題の研究 (2011)
  • Author(s): Hiroki Takahasi
  • Organizer: Dynamics and Complex Systems
  • Place of Presentation: 北海道大学
• [Presentation] Thermodynamic Formalism for the Henon Maps (2011)
  • Author(s): Hiroki Takahasi
  • Organizer: Frontiers in Dynamical Systems and Topology
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] カオス的な２次写像の大偏差原理 (2011)
  • Author(s): Hiroki Takahasi
  • Organizer: エルゴード理論研究集会
  • Place of Presentation: 大阪大学
• [Presentation] Prevalence of nonuniform hyperbolicity at the first bifurcation of the Henon family
  • Author(s): Hiroki Takahasi
  • Organizer: Pacific Rim Conference on Mathematics
  • Place of Presentation: 札幌コンベンションセンター
• [Remarks] List of Papers(Hiroki Takahasi)
  • URL: http://www.math.keio.ac.jp/~hiroki/papers.html