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Goldberg-Milner conjecture and degenerate Beltrami equations in complex dynamics

Research Project

Project/Area Number 16K05193
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionTokyo Institute of Technology

Principal Investigator

Kawahira Tomoki  東京工業大学, 理学院, 准教授 (50377975)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords複素力学系 / ジュリア集合 / 構造安定性 / 放物的分岐 / Beltrami方程式 / 擬等角写像 / 正則力学系 / 反正則力学系 / マンデルブロー集合 / ジュリア集合の / Zalcmanの補題 / Julia集合 / 正則運動 / 関数論 / 複素解析
Outline of Final Research Achievements

In complex dynamics, the periodic points with multiplicity are called parabolic periodic points. If one perturb the mapping that generates the dynamical system, parabolic cycles bifurcates and the dynamics exhibits drastic change. mics totally different. In 1990s, Goldberg and Milnor conjectured that one can choose an appropriate direction of the perturbation in such a way that the dynamics changes quite tamely. In this research program, I take an analytic approach to this conjecture by using degenerate Beltrami equations, and obtained an alternative way to prove some known results about this conjecture.

Academic Significance and Societal Importance of the Research Achievements

複素力学系とは複素数でパラメーター付けされる空間において,時間発展するシステムである.そのような力学系はほとんどの場合安定であることが期待されるが,研究としてはむしろ,力学系を「不安定」にする要因を特定する過程が面白い.本研究では,そのような不安定性要因として代表的なものである「放物的分岐」という現象をコントロールする方法として,新たに解析的なアプローチを提案した.

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (31 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (4 results) Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (17 results) (of which Int'l Joint Research: 14 results,  Invited: 14 results) Book (1 results) Remarks (5 results)

  • [Int'l Joint Research] 中央研究院(その他の国・地域(台湾))

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] 中央研究院(台湾)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Academia Sinica(台湾)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Academia Sinca(台湾)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn2020

    • Author(s)
      Tomoki Kawahira
    • Journal Title

      Anal. Math. Phys.

      Volume: 10

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Simple proofs for the derivative estimates of the holomorphic motion near two boundary points of the Mandelbrot set2019

    • Author(s)
      Yi-Chiuan Chen and Tomoki Kawahira
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 473 Issue: 1 Pages: 345-356

    • DOI

      10.1016/j.jmaa.2018.12.052

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] From Cantor to Semi-hyperbolic Parameter along External Rays2019

    • Author(s)
      Yi-Chiuan Chen and Tomoki Kawahira
    • Journal Title

      To appear in Transactions of the American Mathematical Society

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The Riemann Hypothesis and Holomorphic Index in Complex Dynamics2016

    • Author(s)
      Tomoki Kawahira
    • Journal Title

      Experimental Mathematics

      Volume: 印刷中 Issue: 1 Pages: 37-46

    • DOI

      10.1080/10586458.2016.1217443

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Derivatives of mildly degenerating holomorphic motions of the quadratic Julia sets\\2019

    • Author(s)
      Tomoki Kawahira
    • Organizer
      NCTS Workshop on Dynamical Systems
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Beltrami equations and its application to complex dynamics2019

    • Author(s)
      Tomoki Kawahira
    • Organizer
      The 6th Uppsala University -- Tokyo Tech Joint Symposium
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] The Riemann Hypothesis and Holomorphic Index in Complex Dynamics2019

    • Author(s)
      Tomoki Kawahira
    • Organizer
      TMS meeting at Taichung
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Riemann hypothesis and complex dynamics.2019

    • Author(s)
      Tomoki Kawahira
    • Organizer
      Value distribution of zeta and L-functions and related topics
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Almost affine copies of the Julia sets in the Mandelbrot set2018

    • Author(s)
      Tomoki Kawahira
    • Organizer
      RIMS研究集会「複素力学系研究とその発展」
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Almost affine copies of the Julia sets in the Mandelbrot set2018

    • Author(s)
      Tomoki Kawahira
    • Organizer
      The Boston-Keio Workshop 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On dynamical and parametric Zalcman functions2018

    • Author(s)
      Tomoki Kawahira
    • Organizer
      Perspectives in Modern Analysis International Conference in honor of Dov Aharonov, Samuel Krushkal, Simeon Reich, and Lawrence Zalcman
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On dynamical and parametric Zalcman functions2017

    • Author(s)
      Tomoki Kawahira
    • Organizer
      平成28年度等角写像論・値分布論研究集会
    • Place of Presentation
      千葉大学
    • Year and Date
      2017-01-22
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Almost conformal copies of the Julia sets in the Mandelbrot set2017

    • Author(s)
      Tomoki Kawahira
    • Organizer
      NTCS Workshop on Dynamical Systems 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On Dynamical and Parametric Zalcman Functions2017

    • Author(s)
      Tomoki Kawahira
    • Organizer
      CMTF 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] The Riemann hypothesis and holomorphic index in complex dynamics2017

    • Author(s)
      Tomoki Kawahira
    • Organizer
      Residues, Dynamics and Hyperfunctions
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Almost conformal copies of the Julia sets in the Mandelbrot set2017

    • Author(s)
      Tomoki Kawahira
    • Organizer
      2017年度日本数学会秋季総合分科会・函数論分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] On Dynamical and Parametric Zalcman Functions2017

    • Author(s)
      Tomoki Kawahira
    • Organizer
      RIMS Workshop on Complex Dynamics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the derivatives of degenerating holomorphic motions of hyperbolic quadratic Julia sets2016

    • Author(s)
      Tomoki Kawahira
    • 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
    • Year and Date
      2016-12-14
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Riemann hypothesis and holomorphic index in complex dynamics2016

    • Author(s)
      Tomoki Kawahira
    • Organizer
      函数論シンポジウム
    • Place of Presentation
      静岡県男女共同参画センター あざれあ
    • Year and Date
      2016-10-09
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Similarity between the Mandelbrot set and the Julia sets2016

    • Author(s)
      Tomoki Kawahira
    • Organizer
      The 3rd UU-TT Symposium
    • Place of Presentation
      Uppsala university
    • Year and Date
      2016-09-13
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] From Cantor to Misiurewicz along parameter ray2016

    • Author(s)
      Tomoki Kawahira
    • Organizer
      2016 NCTS Workshop on Dynamical Systems and 2016
    • Place of Presentation
      国立台湾大学
    • Year and Date
      2016-08-15
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 入門複素関数2019

    • Author(s)
      川平友規
    • Total Pages
      233
    • Publisher
      裳華房
    • ISBN
      4785315792
    • Related Report
      2018 Research-status Report
  • [Remarks] 川平友規のweb page

    • URL

      http://www.math.titech.ac.jp/~kawahira

    • Related Report
      2019 Annual Research Report
  • [Remarks] 研究者webページ

    • URL

      http://www.math.titech.ac.jp/~kawahira

    • Related Report
      2018 Research-status Report
  • [Remarks] Beltrami方程式勉強会のページ

    • URL

      https://sites.google.com/view/beltrami-part-ii

    • Related Report
      2018 Research-status Report
  • [Remarks] 研究代表者(個人webページ)

    • URL

      http://www.math.titech.ac.jp/~kawahira

    • Related Report
      2017 Research-status Report
  • [Remarks] 研究者ホームページ

    • URL

      http://www.math.titech.ac.jp/~kawahira/index.html

    • Related Report
      2016 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2022-08-22  

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