Research of rational semigroups, complex dynamics on fiber budles, and random complex dynamics

• Project/Area Number: 18540216
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Osaka University
• Principal Investigator: SUMI Hiroki Osaka University, 大学院・理学研究科, 准教授 (40313324)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥4,060,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥660,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
• Keywords: 解析学 / 複素力学系 / カオス / フラクタル / ランダムな力学系 / 特異関数 / ランダムな複素力学系

Research Abstract

有理関数や多項式の族により生成された半群のリーマン球面上での力学系と、ランダムな複素力学系、ジュリア集合などのフラクタル図形に関する幾何学、などについて、それらを互いに交錯させながら基礎から理論を構築した。このことにより、一つの正則写像を反復合成させる従来の複素力学系では現れないような興味深い現象を数多く発見した。そしてこれらの現象の背景とメカニズムを深く研究した。

Research Products

• [Journal Article] Analytic families of holomorphic iterated function systems (2008)
  • Author(s): M. Roy, H. Sumi and M. Urbanski
  • Journal Title: Nonlinearity 21 Pages: 2255-2279
  • Peer Reviewed
• [Journal Article] Analytic families of holomorphic iterated functionsystems (2008)
  • Author(s): Mario Roy, Hiroki Sumi, Mariusz Urbanski
  • Journal Title: Nonlinearity 21 Pages: 2255-2279
  • Peer Reviewed
• [Journal Article] Random dynamics of polynomials and devil's-staircase-like functions in the complex plane (2007)
  • Author(s): H. Sumi
  • Journal Title: Appl. Math. Comput 187, no.1 Pages: 489-500
• [Journal Article] Structure of Julia sets of polynomial semigroups with bounded finite postcritical set (2007)
  • Author(s): R. Stankewitz and H. Sumi
  • Journal Title: Appl. Math. Comput 187, no.1 Pages: 479-488
• [Journal Article] Random Julia sets that are Jordan curves but not quasicircles (2007)
  • Author(s): H. Sumi
  • Journal Title: 数理解析研究所講究録1537「複素力学系とその周辺」 Pages: 1-32
• [Journal Article] Random dynamics of polynomials and devil's-staircase-like functions in the complex plane (2007)
  • Author(s): H. Sumi
  • Journal Title: Appl. Math. Comput. 187 Pages: 489-500
• [Journal Article] Structure of Julia sets of polynomial semigroups with bounded finite postcritical set (2007)
  • Author(s): R. Stankewitz and H. Sumi
  • Journal Title: Appl. Math. Comput. 187 Pages: 479-488
• [Journal Article] The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity (2006)
  • Author(s): H. Sumi
  • Journal Title: 数理解析研究所講究録 1494 Pages: 62-86
• [Journal Article] Semi-hyperbolic fibered rational maps and rational semigroups (2006)
  • Author(s): H. Sumi
  • Journal Title: Ergod. Th. and Dynam. Sys {\bf 26} Pages: 893-922
  • Peer Reviewed
• [Journal Article] Semi-hyperbolic fibered rational maps and rational semigroups (2006)
  • Author(s): 角 大輝
  • Journal Title: Ergodic Theory and Dynamical Systems 26 Pages: 893-922
• [Journal Article] The equilibrium states for semigroups of rational maps
  • Author(s): H. Sumi and M. Urbanski
  • Journal Title: Monatsh. Math. (掲載予定)
  • Peer Reviewed
• [Journal Article] Real analyticity of Hausdorff dimension for expanding semigroups
  • Author(s): H. Sumi and M. Urbanski
  • Journal Title: Ergod. Th. and Dynam. Sys (掲載予定)
  • Peer Reviewed
• [Journal Article] 有理半群, ランダムな複素力学系と複素平面上の特異関数
  • Author(s): 角大輝
  • Journal Title: 日本数学会雑誌「数学」論説 (2009年4月号掲載予定)
  • NAID: 130004558891
  • Peer Reviewed
• [Presentation] Random complex dynamics and singular functions on the complex plane (2009)
  • Author(s): H. Sumi
  • Organizer: UK-Japan Winter School Dynamics and Complexity
  • Place of Presentation: University of Bath (Bath, イギリス)
  • Year and Date: 2009-01-08
• [Presentation] The space of postcritically bounded 2-generator polynomial semigroupswith hyperbolicity」 (2008)
  • Author(s): H. Sumi
  • Organizer: Moduli and invariants in complex analysis and algebraic geometry
  • Year and Date: 2008-11-27
• [Presentation] Random complex dynamics and singular functions on the complex plane (2008)
  • Author(s): H. Sumi
  • Organizer: Complex dynamics and related topics
  • Place of Presentation: Fudan University(上海、中国)
  • Year and Date: 2008-11-01
• [Presentation] Random complex dynamics and singular functions on thecomplex plane (2008)
  • Author(s): Hiroki Sumi
  • Organizer: Aspects of Transcendental Dynamics
  • Place of Presentation: Jacobs University Bremen (ドイツ)
  • Year and Date: 2008-06-19
• [Presentation] Random complex dynamics and singular functions on the complex plane (2008)
  • Author(s): H. Sumi
  • Organizer: Aspects of Transcendental Dynamics
  • Place of Presentation: Jacobs University Bremen (ドイツ)
  • Year and Date: 2008-06-18
• [Presentation] Random complex dynamics and semigroups of holomorphic maps (2008)
  • Author(s): 角大輝
  • Organizer: 2007年度冬の力学系研究集会
  • Place of Presentation: 日本大学軽井沢研修所
• [Presentation] ランダムな複素力学系、有理半群と複素平面上の特異関数 (2007)
  • Author(s): 角大輝
  • Organizer: 力学系と微分方程式
  • Place of Presentation: 広島大学
  • Year and Date: 2007-11-09
• [Presentation] Random dynamics of polynomials and singular functions in the complex plane (2007)
  • Author(s): H. Sumi
  • Organizer: AMS meeting Special session Complex Dynamics and Complex Function Theory
  • Place of Presentation: Miami University (アメリカ)
  • Year and Date: 2007-03-16
• [Presentation] Random Julia sets that are Jordan curves but not quasicircles (2007)
  • Author(s): 角大輝
  • Organizer: 力学系研究集会
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2007-01-06
• [Presentation] Random complex dynamics and rational semigroups (2007)
  • Author(s): 角大輝
  • Organizer: 複素力学系とその周辺分野の研究
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Classification of polynomial semigroups and random Julia sets that are Jordan curves but not quasicircles (2007)
  • Author(s): H. Sumi
  • Organizer: 15th International conference of finite and infinite dimensional complex analysis and applications
  • Place of Presentation: 大阪市立大学
• [Presentation] Random Julia sets that are Jordan curves but not quasicircles (2006)
  • Author(s): 角大輝
  • Organizer: 複素力学系とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2006-10-02
• [Presentation] 有理半群、ランダムな複素力学系と複素平面上の特異関数 (2006)
  • Author(s): 角大輝
  • Organizer: 日本数学会 函数論分科会特別講演
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2006-09-20
• [Presentation] Dynamics of polynomial semigroups, interaction cohomology and singular functions in the complex plane (2006)
  • Author(s): 角大輝
  • Organizer: 第53回トポロジーシンポジウム
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2006-08-08
• [Presentation] Random dynamics of polynomials and singular functions in the complex plane (2006)
  • Author(s): 角大輝
  • Organizer: Mathematics on Fractals 2006
  • Place of Presentation: 京都大学
• [Presentation] Random dynamics of polynomials and devil's-staircase-like functions in the complex plane (2005)
  • Author(s): H. Sumi
  • Organizer: INTERNATIONAL SYMPOSIUM Analytic Function Theory, Fractional Calculus and Their Applications
  • Place of Presentation: University of Victoria (カナダ)
• [Remarks]
  • URL: http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-j.html
• [Remarks]
  • URL: http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-j.html