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The structure of chaotic regions for area-preserving maps

Research Project

Project/Area Number 15K13444
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

Shishikura Mitsuhiro  京都大学, 理学研究科, 教授 (70192606)

Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords力学系 / カオス / 分岐 / フラクタル / くりこみ / 面積保存
Outline of Final Research Achievements

As a first step toward the study of area preserving 2-dimensional maps, jointly with D. Marti Pete, we studied 1-dimensional standard maps (Arnold family) z-> z+α+β sin(2πz) with complex phase variable z and complex parameters α, β. When real positive β is fixed and α is taken as the main parameter, finger-like structures have been observed near rational α in the parameter space. We used the theory of parabolic bifurcation to analyze this structure. The translation parameter in the Fatou coordinate is a new control parameter in which finger-like structure become a simple horizontal strip structure and this explains the fingers.
We also studied the family of real quadratic polynomials and introduced a new approach using Yoccoz puzzles to show that chaotic parameters have positive measure(Jakobson's theorem). This method together with complex analytic method to estimate the modulus of certain annuli gave a quantitative estimate of the parameter measure.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (17 results)

All 2018 2017 2016 2015

All Journal Article (1 results) (of which Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (16 results) (of which Int'l Joint Research: 15 results,  Invited: 16 results)

  • [Journal Article] Conformality of quasiconformal mappings at a point, revisited2018

    • Author(s)
      Mitsuhiro Shishikura
    • Journal Title

      Annales Academae Scientiarum Fennicae

      Volume: 出版予定

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Presentation] Jakobson's Theorem via Yoccoz puzzles2018

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      On geometric complexity of Julia sets
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Realization problem for tropical complex dynamics2017

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Complex Dynamics: Iterations, Foliations and Evolutions
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Fingers in complex Arnold family2017

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Workshop on New Frontiers in Complex Dynamics: From One to Several Variables
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Matings and Thurston obstruction2017

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Complex dynamics and quasi-conformal geometry
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / 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 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Fractal nature of Julia sets2016

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      First Hong-Kong/Kyoto Workshop on Fractal and related areas
    • Place of Presentation
      The Hong Kong University od Science and Technology, Hong, Kong, China
    • Year and Date
      2016-03-21
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Tropical Complex Dynamics2016

    • Author(s)
      宍倉光広
    • Organizer
      Taking the Measure of One-Dimensional Dynamics
    • Place of Presentation
      Lorentz Center, Leiden, Netherlands
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Tropical Complex Dynamics2016

    • Author(s)
      宍倉光広
    • Organizer
      Mexican-American Workshop in Holomorphic Dynamics
    • Place of Presentation
      Fiesta Americana Condesa hotel in Cancun, Mexico
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Tropical Complex Dynamics2016

    • Author(s)
      宍倉光広
    • Organizer
      Dynamical Systems Seminar, PUC, Chile
    • Place of Presentation
      Pontificia Universidad Catolica de Chile, Santiago, Chile
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Tropical limit of complex dynamical systems2016

    • Author(s)
      宍倉光広
    • Organizer
      Analysis and Dynamics
    • Place of Presentation
      Casa Matematica Oaxaca, Mexico
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rempe model and Denjoy odometer for hedgehogs of complex quadratic polynomials2015

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Conference of Complex Analysis in China 2015
    • Place of Presentation
      Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
    • Year and Date
      2015-10-12
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Complex Dynamics and Teichmueller spaces2015

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      The 23rd International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
    • Place of Presentation
      九州産業大学(福岡県・福岡市)
    • Year and Date
      2015-08-27
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rempe model and Denjoy odometer for hedgehogs of complex quadratic polynomials2015

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      Dynamical Developments: a conference in Complex Dynamics and Teichmueller theory
    • Place of Presentation
      Jacobs University, Bremen, Germany
    • Year and Date
      2015-08-21
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rempe model and Denjoy odometer for hedgehogs of complex quadratic polynomials2015

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      School and Conference on Dynamical Systems
    • Place of Presentation
      Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
    • Year and Date
      2015-08-04
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Smoothness of hairs for some entire functions2015

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      AMS-EMS-SPM International Meeting in Portugal
    • Place of Presentation
      University of Porto, Portugal
    • Year and Date
      2015-06-10
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Toward arithmetic surgery of rational maps2015

    • Author(s)
      Mitsuhiro Shishikura
    • Organizer
      IMSXXV ― Celebrating 25 years of low-dimensional dynamics at Stony Brook
    • Place of Presentation
      Stony Brook University, USA
    • Year and Date
      2015-05-08
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2015-04-16   Modified: 2019-03-29  

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