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Dynamics of transcendental and rational functions

Research Project

Project/Area Number 16F16807
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

宍倉 光広  京都大学, 理学研究科, 教授 (70192606)

Co-Investigator(Kenkyū-buntansha) MARTI-PETE DAVID  京都大学, 理学(系)研究科(研究院), 外国人特別研究員
Project Period (FY) 2016-11-07 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2018: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2017: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2016: ¥300,000 (Direct Cost: ¥300,000)
Keywords力学系 / 超越整関数 / ジュリア集合 / フラクタル / 遊走領域 / 超越関数 / 分岐 / ファトゥー集合
Outline of Annual Research Achievements

C. Bishopは、Eremenko-Lyubich型(class B)超越関数の遊走領域の擬等角折り込みによる構成法を与えた。それによれば、平面上のいくつかの条件をみたす無限樹木から出発して、それに対応する写像を擬正則写像として構成し、それを擬等角写像の合成により、正則な関数に戻すことが出来、もし、その際の歪みをコントロールできるならば、与えられた力学系についての情報を結論できるというものである。David Marti Peteが中心となって宍倉、木坂正史と共にこの論文(そしてそこから派生したLazebnik, Fagella-Godillon-Jarqueの論文も)を読むセミナーを行い、その証明について検討した。これらの論文中の証明の帰納的構成に関する問題点を発見し、検討するうちに、Marti Peteと宍倉は、より簡明な構成法を発見した。我々の方法では、具体的なcosh関数から出発して、一部で写像を擬正則写像として変形し、擬等角折り込みの方法で超越整関数を作るものであるが、各所で具体的な構成を与えているため、Bishop達の証明であった帰納的な構成における循環的な問題は回避できた。この方法により、振動型遊走領域を持つEremenko-Lyubich型超越関数で、さらに有限位数をもつものを構成できることをした。実際に任意の正の半整数がその位数として実現できることも示せた。この結果は論文として投稿中である。
また、宍倉は、上記の構成でも使われる擬等角写像の評価を与え、擬等角写像の1点ごとの微分可能性をより簡明に証明する方法を与えた。
以前から続けている、Marti Peteと宍倉による複素1次元Arnold族のfinger状集合については、fingerの数の定量的評価を与えることを目標とし、論文を準備中である。

Research Progress Status

平成30年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

平成30年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • Research Products

    (22 results)

All 2019 2018 2017 2016

All Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Open Access: 2 results) Presentation (18 results) (of which Int'l Joint Research: 8 results,  Invited: 7 results)

  • [Journal Article] The escaping set of transcendental self-maps of the punctured plane2018

    • Author(s)
      MARTI-PETE DAVID
    • Journal Title

      Ergodic Theory and Dynamical Systems

      Volume: 38 Issue: 2 Pages: 739-760

    • DOI

      10.1017/etds.2016.36

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Conformality of quasiconformal mappings at a point, revisited.2018

    • Author(s)
      M. Shishikura
    • Journal Title

      Ann. Acad. Sci. Fenn. Math.

      Volume: 43 Pages: 981-990

    • DOI

      10.5186/aasfm.2018.4359

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The escaping set of transcendental self-maps of the punctured plane2018

    • Author(s)
      Marti-Pete, David
    • Journal Title

      Ergodic Theory and Dynam. Systems

      Volume: 38 Pages: 739-760

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Dynamic rays of bounded-type transcendental self-maps of the punctured plane2017

    • Author(s)
      Fagella, Nu;ria; Marti-Pete, David
    • Journal Title

      Discrete Contin. Dyn. Syst.

      Volume: 37 Pages: 3123-3160

    • Related Report
      2017 Annual Research Report 2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Wandering domains for entire functions of finite order in the class B2019

    • Author(s)
      David, Marti Pete
    • Organizer
      Postgraduate Conference in Complex Dynamics 2019
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Wandering domains for entire functions of finite order in the class B2019

    • Author(s)
      David, Marti Pete
    • Organizer
      Topics in Complex Dynamics 2019
    • Related Report
      2018 Annual Research Report
  • [Presentation] Oscillating wandering domains for entire functions of finite order in the class B2018

    • Author(s)
      David, Marti Pete
    • Organizer
      New Developments in Complex Analysis and Function Theory
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Wandering domains for entire functions of finite order in the class B2018

    • Author(s)
      David, Marti Pete
    • Organizer
      Complex Dynamics Working Seminar
    • Related Report
      2018 Annual Research Report
  • [Presentation] Quasiconformal variation of cross-ratios and applications2018

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Perspectives in Modern Analysis
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quasiconformal variation of cross-ratios and applications2018

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      New Developments in Complex Analysis and Function Theory
    • Related Report
      2018 Annual Research Report
  • [Presentation] Oscillating wandering domain for a transcendental entire function of class B2018

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Resonances of complex dynamics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The escaping set of transcendental self-maps of the punctured plane2018

    • Author(s)
      Marti-Pete, David
    • Organizer
      数理科学セミナー, 高知大学
    • Related Report
      2017 Annual Research Report
  • [Presentation] Fingers in the parameter space of the complex standard family2018

    • Author(s)
      Marti-Pete, David
    • Organizer
      2018年度 冬の力学系研究集会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Fingers in the parameter space of the complex standard family2017

    • Author(s)
      Marti-Pete, David
    • Organizer
      RIMS Workshop on Complex Dynamics
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Fingers in the parameter space of the complex standard family2017

    • Author(s)
      Marti-Pete, David
    • Organizer
      Dynamical Systems Seminar, Liverpool University,
    • Related Report
      2017 Annual Research Report
  • [Presentation] Fingers in the parameter space of the complex standard family2017

    • Author(s)
      Marti-Pete, David
    • Organizer
      Pure and Applied Mathematics Colloquium, The Open University, UK
    • Related Report
      2017 Annual Research Report
  • [Presentation] Fingers in the parameter space of the complex standard family2017

    • Author(s)
      Marti-Pete, David
    • Organizer
      Workshop “Topics in Complex Dynamics 2017, Universitat de Barcelona
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Fingers in the parameter space of the complex standard family2017

    • Author(s)
      Marti-Pete, David
    • Organizer
      Complex Dynamics Working Seminar Universitat de Barcelona
    • Related Report
      2017 Annual Research Report
  • [Presentation] The escaping set of transcendental self-maps of the punctured plane2017

    • Author(s)
      Marti-Pete, David
    • Organizer
      Residues, dynamics and hyperfunctions, 北海道大学
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] The escaping set of transcendental self-maps of the punctured plane2017

    • Author(s)
      David Marti Pete
    • Organizer
      京都力学系セミナー
    • Place of Presentation
      京都大学理学研究科
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Jakobson's theorem via Yoccoz puzzles and the measure of stochastic parameters2017

    • Author(s)
      宍倉光広
    • Organizer
      Kanto Dynamics Days 2017
    • Place of Presentation
      東京大学数理科学研究科
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The escaping set of transcendental self-maps of the punctured plane2016

    • Author(s)
      David Marti Pete
    • Organizer
      複素力学系およびそのモヂュライ等の関連分野の研究
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited

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Published: 2016-11-08   Modified: 2024-03-26  

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