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Research on semigroups of holomorphic maps and random complex dynamics

Research Project

Project/Area Number 21540216
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionOsaka University

Principal Investigator

SUMI Hiroki  大阪大学, 大学院・理学研究科, 准教授 (40313324)

Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords複素力学系 / カオス / フラクタル幾何学 / ランダム力学系 / 複素解析学 / エルゴード理論 / ハウスドルフ次元 / 雑音誘起現象 / フラクタル / 解析学 / 複素解析 / ランダム複素力学系 / 協調原理 / 有理半群 / 複素特異関数 / 確率複素力学系
Research Abstract

I investigated the dynamics of semigroups of holomorphic maps on the Riemann sphere and the random complex dynamics on the Riemann sphere. I found many new phenomena which cannot hold in the usual iteration dynamics of a single holomorphic map. Moreover, I investigated those phenomena systematically. In particular,
I found that in random complex dynamics, for a generic system, the chaos of the averaged system disappears due to the automatic cooperation of many kinds of maps in the system(cooperation principle). Also, I investigated many complex phenomena of limit functions of the systems. Moreover, I investigated stability and bifurcation of the systems.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (66 results)

All 2012 2011 2010 2009 Other

All Journal Article (21 results) (of which Peer Reviewed: 20 results) Presentation (41 results) Remarks (4 results)

  • [Journal Article] Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups2011

    • Author(s)
      R. Stankewitz, H. Sumi
    • Journal Title

      Trans. Amer. Math. Soc

      Volume: no.10 Issue: 10 Pages: 5293-5319

    • DOI

      10.1090/s0002-9947-2011-05199-8

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups2011

    • Author(s)
      H. Sumi, M. Urbanski
    • Journal Title

      Discrete and Continuous Dynamical Systems Series A

      Volume: Vol.30, No.1 Issue: 1 Pages: 313-363

    • DOI

      10.3934/dcds.2011.30.313

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Dynamics of postcritically bounded polynomial semigroups I : connected components of the Julia sets2011

    • Author(s)
      H. Sumi
    • Journal Title

      Discrete and Continuous Dynamical Systems Series A

      Volume: Vol.29, No.3 Issue: 3 Pages: 1205-1244

    • DOI

      10.3934/dcds.2011.29.1205

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Random complex dynamics and semigroups of holomorphic maps2011

    • Author(s)
      H. Sumi
    • Journal Title

      Proc. London. Math. Soc

      Volume: 102(1) Issue: 1 Pages: 50-112

    • DOI

      10.1112/plms/pdq013

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Random complex dynamics and semigroups of holomorphic maps2011

    • Author(s)
      H.Sumi
    • Journal Title

      Proc.London.Math.Soc.

      Volume: 102 Pages: 50-112

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Dynamics of postcritically bounded polynomial semigroups I : connected comnonents of the Julia sets2011

    • Author(s)
      H.Sumi
    • Journal Title

      Discrete and Continuous Dynamical Systems Series A

      Volume: 29 Pages: 1205-1244

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Dynamics of postcritically bounded polynomial semigroups III : classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles2011

    • Author(s)
      H.Sumi
    • Journal Title

      Ergodic Theory Dynam.Systems

      Volume: 30 Pages: 1869-1902

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Dynamics of postcritically bounded polynomial semigroups III : classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles2010

    • Author(s)
      H. Sumi
    • Journal Title

      Ergodic Theory Dynam. Systems

      Volume: 30, No.6 Pages: 1869-1902

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Rational semigroups, random complex dynamics and singular functions on the complex plane, Selected Papers on Analysis and Differential Equations2010

    • Author(s)
      H. Sumi
    • Journal Title

      Amer. Math. Soc. Transl

      Volume: Vol.230 Pages: 161-200

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Real analyticity of Hausdorff dimension for expanding rational semigroups2010

    • Author(s)
      H. Sumi, M. Urbanski
    • Journal Title

      Ergodic Theory Dynam. Systems

      Volume: Vol.30, No.2 Pages: 601-633

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Rational semigroups, random complex dynamics and singular functions on the complex plane2010

    • Author(s)
      H.Sumi
    • Journal Title

      Amer.Math.Soc.Transl.(2)

      Volume: 230 Pages: 161-200

    • NAID

      130004558891

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Real analyticity of Hausdorff dimension for expanding rational semigroups2010

    • Author(s)
      H.Sumi, M.Urbanski
    • Journal Title

      Ergodic Theory Dynam.Systems

      Volume: 30 Pages: 601-633

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Interaction cohomology of forward or backward self-similar systems2009

    • Author(s)
      H. Sumi
    • Journal Title

      Adv. Math

      Volume: 222(3) Pages: 729-781

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The equilibrium states for semigroups of rational maps2009

    • Author(s)
      H. Sumi, M. Urbanski
    • Journal Title

      Monatsh. Math

      Volume: no.4 Pages: 371-390

    • Related Report
      2011 Final Research Report
    • 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2009

    • Author(s)
      角大輝
    • Journal Title

      雑誌「数学」

      Volume: 第61卷第2号 Pages: 133-161

    • NAID

      130004558891

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Interaction cohomology of forward or backward self-similar systems2009

    • Author(s)
      角大輝
    • Journal Title

      Advances in Mathematics 222

      Pages: 729-781

    • Related Report
      2009 Annual Research Report
    • 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2009

    • Author(s)
      角大輝
    • Journal Title

      数学 61

      Pages: 133-161

    • NAID

      130004558891

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lambda-tolology VS. pointwise topology2009

    • Author(s)
      M.Roy, H.Sumi, M.Urbanski
    • Journal Title

      Ergodic Theory Dynam. Systems 29

      Pages: 685-713

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The equilibrium states for semigroups of rational maps2009

    • Author(s)
      H.Sumi, M.Urbanski
    • Journal Title

      Monatsh.Math. 156

      Pages: 371-390

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article]2009

    • Author(s)
      数学書房編集部編
    • Journal Title

      この定理が美しい(数学書房)

      Pages: 108-117

    • Related Report
      2009 Annual Research Report
  • [Journal Article] Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

    • Author(s)
      H. Sumi, M. Urbanski
    • Journal Title

      Discrete and Continuous Dynamical Systems Ser

      Volume: (掲載予定)

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Presentation] タイトル「Randomness-induced phenomena in random complex dynamics」2012

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「Dynamics of Complex Systems」
    • Place of Presentation
      北海道大学理学部
    • Year and Date
      2012-03-08
    • Related Report
      2011 Final Research Report
  • [Presentation] Randomness-induced phenomena in random complex dynamics2012

    • Author(s)
      Hiroki Sumi
    • Organizer
      Dynamics of Complex Systems
    • Place of Presentation
      北海道大学理学部(北海道)(招待講演)
    • Year and Date
      2012-03-08
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Stability, Bifurcation and Classification of Minimal Sets in Random Complex Dynamics」2012

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「複素力学系の総合的研究」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2012-01-27
    • Related Report
      2011 Final Research Report
  • [Presentation] Stability, Bifurcation and Classification of Minimal Sets in Random Complex Dynamics2012

    • Author(s)
      Hiroki Sumi
    • Organizer
      複素力学系の総合的研究
    • Place of Presentation
      京都大学数理解析研究所(京都府)
    • Year and Date
      2012-01-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Stability and bifurcation in random complex dynamics」2012

    • Author(s)
      H. Sumi
    • Organizer
      mini-workshop`` Thermodynamic Formalism, Geometry and Stochastics
    • Place of Presentation
      ドイツOberwolfach
    • Year and Date
      2012-01-02
    • Related Report
      2011 Final Research Report
  • [Presentation] Stability and bifurcation in random complex dynamics2012

    • Author(s)
      Hiroki Sumi
    • Organizer
      Mini-workshop "Thermodynamic Formalisms, Geometry and Stochastics"
    • Place of Presentation
      ドイツOberwolfach(招待講演)
    • Year and Date
      2012-01-02
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Cooperation principle and stability in random complex dynamics」2011

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「Automorphisms of algebraic varieties-Dynamics and Arithmetic」
    • Place of Presentation
      和歌山県白浜町
    • Year and Date
      2011-12-19
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle and stability in random complex dynamics2011

    • Author(s)
      Hiroki Sumi
    • Organizer
      Automorphisms of algebraic varieties-Dynamics and Arithmetic
    • Place of Presentation
      ラフォーレ南紀白浜(和歌山県)(招待講演)
    • Year and Date
      2011-12-19
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Stability and bifurcation in random complex dynamics」2011

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「The 19th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications」
    • Place of Presentation
      広島市アステールプラザ
    • Year and Date
      2011-12-12
    • Related Report
      2011 Final Research Report
  • [Presentation] Stability and bifurcation in random complex dynamics2011

    • Author(s)
      Hiroki Sumi
    • Organizer
      The 19^<th> International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
    • Place of Presentation
      広島市アステールプラザ(広島県)
    • Year and Date
      2011-12-12
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Stability and bifurcation in random complex dynamics」2011

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「エルゴード理論とその周辺」
    • Place of Presentation
      大阪大学
    • Year and Date
      2011-12-08
    • Related Report
      2011 Final Research Report
  • [Presentation] Stability and bifurcation in random complex dynamics2011

    • Author(s)
      Hiroki Sumi
    • Organizer
      エルゴート理論とその周辺
    • Place of Presentation
      大阪大学(大阪府)
    • Year and Date
      2011-12-08
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Cooperation principle and stability in random complex dynamics」2011

    • Author(s)
      H. Sumi
    • Organizer
      ドイツ研究集会Goettingen-Penn State International Summer School on Dynamical Systemsドイツ
    • Place of Presentation
      ゲッチンゲン大学
    • Year and Date
      2011-08-10
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle and disappearance of chaos in random complex dynamics2011

    • Author(s)
      Hiroki Sumi
    • Organizer
      Goettinge-Penn State International Summer School on Dynamical Systems
    • Place of Presentation
      ドイツゲッチンゲン大学(招待講演)
    • Year and Date
      2011-08-10
    • Related Report
      2011 Annual Research Report
  • [Presentation] タイトル「Cooperation principle and stability in random complex dynamics」2011

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「Dynamics of Complex Systems 2011」
    • Place of Presentation
      北海道大学
    • Year and Date
      2011-03-09
    • Related Report
      2011 Final Research Report
  • [Presentation] タイトル「Cooperation principle, stability and bifurcation in random complex dynamics」2011

    • Author(s)
      角大輝
    • Organizer
      2010年度冬の力学系研究集会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2011-01-08
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle, stability and bifurcation in random complex dynamics2011

    • Author(s)
      角大輝
    • Organizer
      2010年度冬の力学系研究集会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2011-01-08
    • Related Report
      2010 Annual Research Report
  • [Presentation] タイトル「Cooperation principle and density of stable systems in random complex dynamics」2010

    • Author(s)
      H. Sumi
    • Organizer
      数理解析研究所研究集会「複素力学系とその周辺分野の研究」
    • Place of Presentation
      京都大学理学部数学教室
    • Year and Date
      2010-12-08
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle and density of stable systems in random complex dynamics2010

    • Author(s)
      角大輝
    • Organizer
      数理解析研究所研究集会 「複素力学系とその周辺分野の研究」
    • Place of Presentation
      京都大学
    • Year and Date
      2010-12-08
    • Related Report
      2010 Annual Research Report
  • [Presentation] タイトル「Cooperation principle, stability and bifurcation in random complex dynamics」2010

    • Author(s)
      角大輝
    • Organizer
      第53回函数論シンポジウム
    • Place of Presentation
      名城大学
    • Year and Date
      2010-11-23
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle, stability and bifurcation in random complex dynamics2010

    • Author(s)
      角大輝
    • Organizer
      第53回 函数論シンポジウム
    • Place of Presentation
      名城大学(愛知県)(招待講演)
    • Year and Date
      2010-11-23
    • Related Report
      2010 Annual Research Report
  • [Presentation] タイトル「Cooperation principle and disappearance of chaos in random complex dynamics」2010

    • Author(s)
      H. Sumi
    • Organizer
      国際研究集会Stochastic Processes and Their Application
    • Place of Presentation
      大阪千里中央ライフサイエンスセンター
    • Year and Date
      2010-09-07
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle and disappearance of chaos in random complex dynamics2010

    • Author(s)
      H.Sumi
    • Organizer
      Stochastic Processes and Their Application
    • Place of Presentation
      大阪 千里中央ライフサイエンスセンター
    • Year and Date
      2010-09-07
    • Related Report
      2010 Annual Research Report
  • [Presentation] Cooperation principle and disappearance of chaos in random complex dynamics2010

    • Author(s)
      H.Sumi
    • Organizer
      Conference on Complex Analysis
    • Place of Presentation
      アメリカ イリノイ大学(招待講演)
    • Year and Date
      2010-05-21
    • Related Report
      2010 Annual Research Report
  • [Presentation] タイトル「Cooperation principle in random complex dynamics and singular functions on the complex plane」2010

    • Author(s)
      角大輝
    • Organizer
      2009年度冬の力学系研究集会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2010-01-10
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle in random complex dynamics and singular functions on the complex plane2010

    • Author(s)
      角大輝
    • Organizer
      2009年度冬の力学系研究集会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2010-01-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] タイトル「Cooperation principle and disappearance of chaos in random complex dynamics」2010

    • Author(s)
      H. Sumi
    • Organizer
      研究集会Conference on Complex Analysis
    • Place of Presentation
      アメリカUniversity of Illinois
    • Related Report
      2011 Final Research Report
  • [Presentation] タイトル「Cooperation principle in random complex dynamics and singular functions on the complex plane」2009

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「複素力学系とその関連分野の総合的研究」
    • Place of Presentation
      京都大学
    • Year and Date
      2009-12-16
    • Related Report
      2011 Final Research Report
  • [Presentation] Cooperation principle in random complex dynamics and singular functions on the complex plane2009

    • Author(s)
      角大輝
    • Organizer
      研究集会「複素力学系とその関連分野の総合的研究」
    • Place of Presentation
      京都大学
    • Year and Date
      2009-12-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] Cooperation principle in random complex dynamics and singular functions on the complex plane2009

    • Author(s)
      角大輝
    • Organizer
      研究集会「エルゴード理論の最近の話題」
    • Place of Presentation
      日本大学 (東京)
    • Year and Date
      2009-11-07
    • Related Report
      2009 Annual Research Report
  • [Presentation] タイトル「ランダムな複素力学系における協調原理と複素平面上の特異関数」2009

    • Author(s)
      角大輝
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2011 Final Research Report
  • [Presentation] ランダムな複素力学系における協調原理と複素平面上の特異関数2009

    • Author(s)
      角大輝
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] タイトル「Random complex dynamics and singular functions on the complex plane」2009

    • Author(s)
      H. Sumi
    • Organizer
      研究集会Kyoto Dynamics Days 8
    • Place of Presentation
      京都大学
    • Year and Date
      2009-09-12
    • Related Report
      2011 Final Research Report
  • [Presentation] Random complex dynamics and singular functions on the complex plane2009

    • Author(s)
      角大輝
    • Organizer
      Kyoto Dynamics Days 8
    • Place of Presentation
      京都大学
    • Year and Date
      2009-09-12
    • Related Report
      2009 Annual Research Report
  • [Presentation] Random complex dynamics and singular functions on the complex plane2009

    • Author(s)
      角大輝
    • Organizer
      Dynamical Sistems II
    • Place of Presentation
      アメリカ・Ball State University
    • Year and Date
      2009-05-17
    • Related Report
      2009 Annual Research Report
  • [Presentation] タイトル「Cooperation principle in random complex dynamics and singular functions on the complex plane」2009

    • Author(s)
      H. Sumi
    • Organizer
      研究集会「エルゴード理論の最近の話題」
    • Place of Presentation
      日本大学
    • Related Report
      2011 Final Research Report
  • [Presentation] タイトル「Cooperation principle in random complex dynamics and singular functions on the complex plane」2009

    • Author(s)
      H. Sumi
    • Organizer
      研究集会XXIst Rolf Nevanlinna Colloquium
    • Place of Presentation
      京都大学
    • Related Report
      2011 Final Research Report
  • [Presentation] タイトル「Random complex dynamics and singular functions on the complex plane」2009

    • Author(s)
      H. Sumi
    • Organizer
      研究集会ADVANCES IN LOW DIMENSIONAL DYNAMICS
    • Place of Presentation
      Stony Brook University(アメリカ)
    • Related Report
      2011 Final Research Report
  • [Presentation] タイトル「Random complex dynamics and singular functions on the complex plane」2009

    • Author(s)
      H. Sumi
    • Organizer
      研究集会Dynamical Sistems II
    • Place of Presentation
      University of North Texas(アメリカ)
    • Related Report
      2011 Final Research Report
  • [Presentation] Random complex dynamics and singular functions on the complex plane2009

    • Author(s)
      角大輝
    • Organizer
      ADVANCES IN LOW DIMENSIONAL DYNAMICS
    • Place of Presentation
      アメリカ・Stony Brook University
    • Related Report
      2009 Annual Research Report
  • [Presentation] Cooperation principle in random complex dynamics and singular functions on the complex plane2009

    • Author(s)
      角大輝
    • Organizer
      XXIst Rolf Nevanlinna Colloquium
    • Place of Presentation
      京都大学
    • Related Report
      2009 Annual Research Report
  • [Remarks] 主な研究論文や雑誌への掲載予定論文、投稿中論文などはホームページ

    • URL

      http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-j.html

    • Related Report
      2011 Final Research Report
  • [Remarks]

    • URL

      http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-j.html

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-j.html

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-j.html

    • Related Report
      2009 Annual Research Report

URL: 

Published: 2009-04-01   Modified: 2016-04-21  

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