thermodynamic formalism and bifurcation of Henon maps

• Project/Area Number: 15H05435
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Keio University
• Principal Investigator: takahasi hiroki 慶應義塾大学, 理工学部(矢上), 准教授 (00467440)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥11,830,000 (Direct Cost: ¥9,100,000、Indirect Cost: ¥2,730,000); Fiscal Year 2018: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2017: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2016: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2015: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
• Keywords: 力学系 / エノン写像 / 大偏差原理 / 可算マルコフシフト / 熱力学形式 / 平衡状態 / 分岐 / エルゴード最適化 / エルゴード理論 / カオス / 双曲性

Outline of Final Research Achievements

I have analyzed the dynamics of Henon maps with the use of the thermodynamic formalism. I have succeeded in showing the existence and uniqueness of equilibrium states, and performed a multifractal analysis on the geometric structure of the invariant set. In 2017, I have started working on the Large deviation principle of countable Markov shifts, an important class of dynamical systems on non-compact spaces. I have estalished the level-2 large deviation principle.

Academic Significance and Societal Importance of the Research Achievements

エノン写像については、最初の分岐パラメータにおいては期待された通りの成果が挙がった。研究開始当初の目標であった分岐パラメータ通過後のダイナミクスの解析については結果を得ることができなかったが、本研究資金による海外研究者との研究交流により研究課題を共有することができ、国際的な共同研究に発展しつつあることには意義がある。大偏差原理の研究については、力学系の観点からの研究がまだそれほど多くないことから、今後の新展開が期待されるだけでなく、確率論など他分野との連携も大いに期待される。

Research Products

• [Int'l Joint Research] University of New Caledonia(フランス)
• [Int'l Joint Research] University of Rochester(米国)
• [Int'l Joint Research] University of Brest(フランス)
• [Int'l Joint Research] University of Rochester(米国)
• [Int'l Joint Research] University of Rochester(米国)
• [Int'l Joint Research] Federal University of Rio de Janeiro(ブラジル)
• [Int'l Joint Research] Universite catholique de Louvain(ベルギー)
• [Int'l Joint Research] ICTP(イタリア)
• [Journal Article] Lyapunov optimization for non-generic one-dimensional expanding Markov maps (2019)
  • Author(s): M. Shinoda, H. Takahasi
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 印刷中 Issue: 9 Pages: 2571-2592
  • DOI: 10.1017/etds.2019.6
  • Peer Reviewed
• [Journal Article] Large deviation principles for countable Markov shifts (2019)
  • Author(s): Hiroki Takahasi
  • Journal Title: Transactions of the American Mathematical Society Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Entropy-approachability for transitive Markov shifts over infinite alphabet (2019)
  • Author(s): Hiroki Takahasi
  • Journal Title: Proceedings of the American Mathematical Society Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Boundary of the horseshoe locus for the Henon family (2018)
  • Author(s): Z. Arai, Y. Ishii, H. Takahasi
  • Journal Title: SIAM Journal on Applied Dynamical Systems Volume: 17 Issue: 3 Pages: 2234-2248
  • DOI: 10.1137/18m1174684
  • Peer Reviewed / Open Access
• [Journal Article] Lyapunov optimization for non-generic one-dimensional expanding Markov maps (2017)
  • Author(s): Mao Shinoda, Hiroki Takahasi
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: -
  • Peer Reviewed
• [Journal Article] Removal of phase transition in the Chebyshev quadratic and thermodynamics for Henon-like maps near the first bifurcation (2016)
  • Author(s): Hiroki Takahasi
  • Journal Title: Journal of Statistical Physics Volume: 164 Issue: 6 Pages: 1354-1378
  • DOI: 10.1007/s10955-016-1584-y
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Equilibrium measures at temperature zero for Henon-like maps at the first bifurcation (2016)
  • Author(s): Hiroki Takahasi
  • Journal Title: SIAM Journal on Applied Dynamical Systems Volume: 15 Issue: 1 Pages: 106-124
  • DOI: 10.1137/15m1015741
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Birkhoff spectrum for Henon-like maps at the first bifurcation (2016)
  • Author(s): Hiroki Takahasi
  • Journal Title: Dynamical Systems: An International Journal Volume: 31 Issue: 1 Pages: 41-59
  • DOI: 10.1080/14689367.2015.1102201
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Lyapunov spectrum for Henon-like maps at the first bifurcation (2016)
  • Author(s): Hiroki Takahasi
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: to appear
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Large deviation principles for countable Markov shifts (2018)
  • Author(s): H Takahasi
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Invited
• [Presentation] Large deviation principle for arithmetic functions in continued fraction expansion (2018)
  • Author(s): H. Takahasi
  • Organizer: ICM Satellite conference on Dynamical Systems and Related Topics
  • Invited
• [Presentation] Large deviation principle for arithmetic mean of digits in continued fraction expansion (2018)
  • Author(s): 高橋博樹
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] 可算マルコフシフトの大偏差原理 (2018)
  • Author(s): 高橋博樹
  • Organizer: 冬の力学系研究集会
• [Presentation] Kyoto Dynamics Days: Random Dynamical Systems Theory and Its Applications (2017)
  • Author(s): 高橋博樹
  • Organizer: Lyapunov optimizing measures of non-generic one-dimensional expanding Markov maps
  • Int'l Joint Research / Invited
• [Presentation] Lyapunov optimizing measures of non-generic one-dimensional expanding Markov maps (2017)
  • Author(s): 高橋博樹
  • Organizer: NCTS Workshop on Dynamical Systems
  • Int'l Joint Research / Invited
• [Presentation] Lyapunov optimizing measures of non-generic one-dimensional expanding Markov maps (2017)
  • Author(s): 高橋博樹
  • Organizer: 日本数学会 秋期総合分科会 統計数学分科会
• [Presentation] non-recurrentでflatな臨界点を持つ区間力学系の大偏差原理とレート関数の零点の構造について (2017)
  • Author(s): 高橋博樹
  • Organizer: 九州力学系セミナー
• [Presentation] Lyapunov optimizing measures of non-generic one-dimensional expanding Markov maps (2017)
  • Author(s): 高橋博樹
  • Organizer: Dynamical Systems Seminar
  • Int'l Joint Research / Invited
• [Presentation] Removal of the phase transition of 1-2x^2 on [-1,1] and the thermodynamics of the Henon map at the first bifurcation (2016)
  • Author(s): 高橋 博樹
  • Organizer: STATPHYS2016
  • Place of Presentation: リヨン（フランス）
  • Year and Date: 2016-07-18
  • Int'l Joint Research
• [Presentation] Removal of phase transition of Chebyshev quadratic and thermodynamics of Henon-like maps near the first bifurcation (2016)
  • Author(s): Hiroki Takahasi
  • Organizer: Fractal Geometry, Hyperbolic Dynamics and Thermodynamic Formalism
  • Place of Presentation: Providence（USA）
  • Year and Date: 2016-03-07
  • Int'l Joint Research
• [Presentation] Removal of the phase transition of 1-2x^2 on [-1,1] and the thermodynamics of the Henon map at the first bifurcation (2016)
  • Author(s): 高橋 博樹
  • Organizer: Probability Seminar
  • Place of Presentation: ロチェスター（アメリカ）
  • Int'l Joint Research / Invited
• [Presentation] Lower estimate of the Hausdorff dimension of the set of zero Lyapunov exponent for non-recurrent unimodal maps with flat critical point (2016)
  • Author(s): 高橋 博樹
  • Organizer: 力学系とその関連分野の連携探索
  • Place of Presentation: 京都大学（京都府・京都市）
• [Presentation] Removal of the phase transition of 1-2x^2 on [-1,1] and the thermodynamics of the Henon map at the first bifurcation (2016)
  • Author(s): 高橋 博樹
  • Organizer: 日本数学会秋期総合分科会（統計数学）
  • Place of Presentation: 関西大学（大阪府・吹田市）
• [Presentation] Flatness-induced phase transition in Lyapunov spectrum for unimodal maps (2016)
  • Author(s): 高橋 博樹
  • Organizer: 九州力学系セミナー
  • Place of Presentation: 九州大学（福岡県・福岡市）
• [Presentation] Flatness-induced phase transition in Lyapunov spectrum for unimodal maps (2016)
  • Author(s): 高橋 博樹
  • Organizer: Dynamics Days Latin America and the Caribbeann
  • Place of Presentation: プエブラ（メキシコ）
  • Int'l Joint Research
• [Presentation] Lyapunov spectum for Henon-like maps at the first bifurcation (2015)
  • Author(s): Hiroki Takahasi
  • Organizer: School and Conference on Dynamical Systems
  • Place of Presentation: Trieste（Italy）
  • Year and Date: 2015-08-04
  • Int'l Joint Research
• [Presentation] ある非一様双曲的1次元写像の逃げ率公式について (2015)
  • Author(s): Hiroki Takahasi
  • Organizer: 力学系とその諸分野への応用
  • Place of Presentation: 京都大学数理解析研究所（京都府・京都市）
  • Year and Date: 2015-07-01
• [Presentation] Escape rate for some non-uniformly hyperbolic one-dimensional map (2015)
  • Author(s): Hiroki Takahasi
  • Organizer: NCTS Workshop on Dynamical Systems
  • Place of Presentation: Taipei（Taiwan）
  • Year and Date: 2015-05-22
  • Int'l Joint Research / Invited
• [Presentation] Large deviation principle for one-dimensional maps under weak hyperbolicity assumptions (2015)
  • Author(s): Hiroki Takahasi
  • Organizer: IMS XXV, Celebrating 25 years of low-dimensional dynamics at Stony Brook University
  • Place of Presentation: Stony Brook（USA）
  • Year and Date: 2015-05-08
  • Int'l Joint Research
• [Remarks] Papers of Hiroki Takahasi