Analytic study of irrationally indifferent cycles in higher dimensional complex dynamics and non-archimedean dynamics

• Project/Area Number: 15K04924
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto Institute of Technology
• Principal Investigator: OKUYAMA YUSUKE 京都工芸繊維大学, 基盤科学系, 教授 (00334954)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 複素力学系 / 非アルキメデス的力学系 / 無理的中立周期系 / 算術力学系

Outline of Final Research Achievements

We studied quantitatively the equidistribution phenomena towards the equilibirium measure and the approximation formula of the Lyapunov exponent with respect to the equilibrium measure from the view points of complex, non-archimedean, and arithmetic dynamics. In arithmetic dynamics, we obtained a quantitative estimate on the approximation of non-constant rational functions by dynamics on a singular domain associated to irrationally indifferent cycles.

Academic Significance and Societal Importance of the Research Achievements

複素力学系や非アルキメデス的力学系においてはそのカオス部分が比較的具体的に解析することが可能であり、本研究はそれを進めてカオス部分を平衡測度を通じて定量的に解析していることに学術的意義があり、一般的なカオス現象の将来の解析の土台ともなり社会的意義もある。

Research Products

• [Int'l Joint Research] Universite de Picardie Jules Verne/Universite de Lille/Ecole Polytechnique(フランス)
• [Int'l Joint Research] Ecole Polytechnique/Universite de Picardie Jules Verne(フランス)
• [Int'l Joint Research] Universite de Picardie Jules Verne(France)
• [Int'l Joint Research] Imperial Colledge London(United KIngdom)
• [Int'l Joint Research] Northwestern University(米国)
• [Journal Article] Geometric formulas on Rumely’s weight function and crucial measure in non-archimedean dynamics (2020)
  • Author(s): Yusuke Okuyama
  • Journal Title: Mathematische Annalen Volume: 376 Issue: 3-4 Pages: 913-956
  • DOI: 10.1007/s00208-019-01884-7
  • Peer Reviewed
• [Journal Article] Hyperbolic components of rational maps: Quantitative equidistribution and counting (2019)
  • Author(s): Thomas Gauthier, Yusuke Okuyama, and Gabriel Vigny
  • Journal Title: Commentarii Mathematici Helvetici Volume: 94 Issue: 2 Pages: 347-398
  • DOI: 10.4171/cmh/462
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A generalization of the converse of Brolin's theorem (2019)
  • Author(s): Yusuke Okuyama and Malgorzata Stawiska
  • Journal Title: Annales Polonici Mathematici Volume: 123 Pages: 467-472
  • DOI: 10.4064/ap180819-23-2
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Discontinuity of a degenerating escape rate (2018)
  • Author(s): Laura DeMarco and Yusuke Okuyama
  • Journal Title: Conformal Geometry and Dynamics Volume: 22 Issue: 3 Pages: 33-44
  • DOI: 10.1090/ecgd/318
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Nevanlinna theory and value distribution in the unicritical polynomials family (2018)
  • Author(s): Yusuke Okuyama
  • Journal Title: Rivista di Matematica della Universita di Parma Volume: 9 Pages: 1-19
  • Peer Reviewed
• [Journal Article] Degeneration of quadratic polynomial endomorphisms to a Henon map (2018)
  • Author(s): Fabrizio Bianchi and Yusuke Okuyama
  • Journal Title: Indiana University Mathematics Journal Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Adelically summable normalized weights and adelic equidistribution of effective divisors having small diagonals and small heights on the Berkovich projective lines (2017)
  • Author(s): Yusuke Okuyama
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B64 Pages: 55-66
  • Peer Reviewed
• [Journal Article] Value distribution of the sequences of the derivatives of iterated polynomials (2017)
  • Author(s): Yusuke Okuyama
  • Journal Title: Annales Academiae Scientiarum Fennicae Mathematica Volume: 42 Pages: 563-574
  • DOI: 10.5186/aasfm.2017.4233
  • Peer Reviewed
• [Journal Article] A Mahler-type estimate of weighted Fekete sums on the Berkovich projective line (2017)
  • Author(s): Yusuke Okuyama
  • Journal Title: Acta Arithmetica Volume: 178 Issue: 3 Pages: 257-272
  • DOI: 10.4064/aa8437-9-2016
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Effective divisors on the projective line having small diagonals and small heights and their application to adelic dynamics (2016)
  • Author(s): Yusuke Okuyama
  • Journal Title: Pacific Journal of Mathematics Volume: 280 Issue: 1 Pages: 141-175
  • DOI: 10.2140/pjm.2016.280.141
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A Lehto--Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space (2015)
  • Author(s): Yusuke Okuyama
  • Journal Title: International Journal of Mathematics Volume: 26 Issue: 06 Pages: 1541009-1541009
  • DOI: 10.1142/s0129167x15410098
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Quantitative approximations of the Lyapunov exponent of a rational function over valued fields (2015)
  • Author(s): Yusuke Okuyama
  • Journal Title: Mathematische Zeitschrift Volume: 280 Issue: 3-4 Pages: 691-706
  • DOI: 10.1007/s00209-015-1443-6
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] A generalization of the converse of Brolin's theorem (2019)
  • Author(s): Yusuke Okuyama
  • Organizer: Geometric Complex Analysis on Foliations and Dynamics
  • Int'l Joint Research / Invited
• [Presentation] An a priori bound of endomorphisms on CPk and its non-archimedean counterpart (2018)
  • Author(s): Yusuke Okuyama
  • Organizer: Complex geometry and complex dynamics in higher dimensions
  • Int'l Joint Research / Invited
• [Presentation] An improvement of McMullen's finiteness theorem (2018)
  • Author(s): Yusuke Okuyama
  • Organizer: Rencontre ANR Fatou
  • Int'l Joint Research / Invited
• [Presentation] Geometric formulas on Rumely’s order function, weight function, and crucial measure in non-archimedean dynamics (2018)
  • Author(s): Yusuke Okuyama
  • Organizer: 代数的整数論とその周辺
  • Int'l Joint Research
• [Presentation] Degeneration of regular quadratic polynomial endomorphisms of C2 towards a Henon map (2018)
  • Author(s): Yusuke Okuyama
  • Organizer: 複素力学系研究とその発展
  • Int'l Joint Research / Invited
• [Presentation] Arithmetic and algebraic geometry of the moduli of complex dynamics (2018)
  • Author(s): Yusuke Okuyama
  • Organizer: The 11th Conference on Arithmetic and Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Discontinuity of the escape rate of a degenerating meromorphic family of rational maps (2017)
  • Author(s): Yusuke Okuyama
  • Organizer: 複素力学系の研究
  • Int'l Joint Research / Invited
• [Presentation] A potential theoretic characterization of polynomials among rational functions on the Berkovich projective line (2016)
  • Author(s): Yusuke Okuyama
  • Organizer: 日本数学会2016年年会函数論分科会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-17
• [Presentation] 複素および非アルキメデス的力学系における多項式のポテンシャル論的特徴付けとその応用 (2016)
  • Author(s): Yusuke Okuyama
  • Organizer: ポテンシャル論とその関連分野
  • Place of Presentation: 北海道大学
  • Year and Date: 2016-01-31
• [Presentation] A Mahler-type estimate of weighted Fekete sums on the Berkovich projective line (2016)
  • Author(s): Yusuke Okuyama
  • Organizer: Journee Amienoise de Systemes Dynamiques 2016
  • Place of Presentation: Universite de Picardie Jules Verne, France
  • Invited
• [Presentation] Quantitative approximation of Lyapunov exponents in non-archimedean and complex dynamics (2016)
  • Author(s): Yusuke Okuyama
  • Organizer: Dynamique holomorphe et non-archimedienne
  • Place of Presentation: Universite de Picardie Jules Verne, France
  • Invited
• [Presentation] A tropical function associated to a rational function (2016)
  • Author(s): Yusuke Okuyama
  • Organizer: Complex dynamical systems and related topics
  • Place of Presentation: RIMS, Kyoto University
  • Int'l Joint Research / Invited
• [Presentation] A characterization of polynomials among rational functions in non-archimedean and complex dy- namics (2015)
  • Author(s): Yusuke Okuyama
  • Organizer: 複素力学系の深化
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-12-10
  • Int'l Joint Research / Invited
• [Presentation] Lehto--Virtanen, Marty, and Zalcman-type theorems and their applications to Kobayashi hyperbolic geometry (2015)
  • Author(s): Yusuke Okuyama
  • Organizer: 日本数学会2015年秋期総合分科会函数論分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-15
• [Presentation] Equidistribution of rational functions having a superattracting periodic point (2015)
  • Author(s): Yusuke Okuyama
  • Organizer: 日本数学会2015年秋期総合分科会函数論分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13