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Analysis of Laplacians on fractals invariant under action of discrete groups of Moebius transformations

Research Project

Project/Area Number 18K18720
Research Category

Grant-in-Aid for Challenging Research (Exploratory)

Allocation TypeMulti-year Fund
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionKyoto University (2021-2023)
Kobe University (2018-2020)

Principal Investigator

KAJINO Naotaka  京都大学, 数理解析研究所, 准教授 (90700352)

Project Period (FY) 2018-06-29 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥6,240,000 (Direct Cost: ¥4,800,000、Indirect Cost: ¥1,440,000)
Fiscal Year 2020: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2019: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2018: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Keywordsラプラシアン / フラクタル / Weyl型固有値漸近挙動 / Klein群の極限集合 / Sierpinski carpet / 自己等角フラクタル曲線 / 複素力学系のJulia集合 / Klein群の擬等角変形 / 擬等角写像
Outline of Final Research Achievements

As the main results of this research project, for certain classes of self-conformal fractals in the plane (fractals in the plane which are self-similar with respect to suitable families of conformal maps), the principal investigator has constructed (candidates of) geometrically canonical Laplacians and revealed detailed properties of them including Weyl type asymptotic behavior of their sequence of eigenvalues.
At the beginning of this research project the class of fractals was limited to (some accessible concrete examples of) circle packing fractals which are invariant under the action of Kleinian groups (discrete groups of Moebius transformations on the complex plane). In the course of the more recent studies, however, the principal investigator has succeeded in extending the same line of results to planar simple fractal curves which are invariant under the action of finite families of general conformal maps (satisfying a certain condition on the non-triviality of the derivatives).

Academic Significance and Societal Importance of the Research Achievements

フラクタル上に自然なラプラシアンを構築し解析する研究は40年近い歴史を有するが,研究対象としては厳密な自己相似性もしくは比較的平易な組み合わせ論的構造を持つフラクタルが主であり,自己等角フラクタルのように一種の自己相似性は確かに持っているものの厳密に自己相似ではないフラクタルに対する研究はごく僅かしかなかった.
本科研費研究課題の結果はそのようなフラクタルにおいて自然な解析学を展開することに成功していると言え,扱えるフラクタルの範疇はまだまだ限定的ではあるものの,既存の研究において取り上げられることの少なかった本質的な問題に一定の理解を与えたその意義は大きい.

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (22 results)

All 2024 2023 2021 2019 2018 Other

All Journal Article (1 results) (of which Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (14 results) (of which Int'l Joint Research: 8 results,  Invited: 10 results) Remarks (6 results) Funded Workshop (1 results)

  • [Journal Article] The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects2021

    • Author(s)
      Naotaka Kajino
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 87 Pages: 293-314

    • DOI

      10.2969/aspm/08710293

    • ISBN
      9784864970952
    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Geometric Laplacians on self-conformal fractal curves in the plane2024

    • Author(s)
      Naotaka Kajino
    • Organizer
      French Japanese Conference on Probability & Interactions
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometric Laplacians on self-conformal fractal curves in the plane2023

    • Author(s)
      Naotaka Kajino
    • Organizer
      Random Interacting Systems, Scaling Limits, and Universality - Week 3: Workshop on Random Interacting Systems
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues2021

    • Author(s)
      Naotaka Kajino
    • Organizer
      Topology and Dynamics Seminar at the University of Birmingham
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues2019

    • Author(s)
      Naotaka Kajino
    • Organizer
      The 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI): Stochastic Analysis, Random Fields and Integrable Probability
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues2019

    • Author(s)
      Naotaka Kajino
    • Organizer
      Research on the Theory of Random Dynamical Systems and Fractal Geometry
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues2019

    • Author(s)
      梶野 直孝
    • Organizer
      第62回函数論シンポジウム
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] The Laplacian on some round Sierpinski carpets and Weyl's asymptotics for its eigenvalues2019

    • Author(s)
      梶野 直孝
    • Organizer
      2018年度冬の力学系研究集会
    • Related Report
      2018 Research-status Report
  • [Presentation] Analysis of diffusions on self-similar and round Sierpinski carpets2019

    • Author(s)
      Naotaka Kajino
    • Organizer
      Okayama Workshop on Stochastic Analysis 2019
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian and Weyl's eigenvalue asymptotics on round Sierpinski carpets realized as the limit sets of certain Kleinian groups2019

    • Author(s)
      梶野 直孝
    • Organizer
      2018年度リーマン面・不連続群論研究集会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] The Laplacian on some round Sierpinski carpets and Weyl's asymptotics for its eigenvalues2019

    • Author(s)
      梶野 直孝
    • Organizer
      2019年度日本数学会年会・幾何学分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] The Laplacian on some round Sierpinski carpets and Weyl's asymptotics for its eigenvalues2018

    • Author(s)
      Naotaka Kajino
    • Organizer
      9th International Conference on Stochastic Analysis and Its Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian on some round Sierpinski carpets and Weyl's asymptotics for its eigenvalues2018

    • Author(s)
      Naotaka Kajino
    • Organizer
      Fractal Geometry and Stochastics 6
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Laplacian on some round Sierpinski carpets and Weyl's asymptotics for its eigenvalues2018

    • Author(s)
      梶野 直孝
    • Organizer
      2018年度エルゴード理論とその周辺
    • Related Report
      2018 Research-status Report
  • [Presentation] The Laplacian on some round Sierpinski carpets and Weyl's asymptotics for its eigenvalues2018

    • Author(s)
      梶野 直孝
    • Organizer
      2018年度確率論シンポジウム
    • Related Report
      2018 Research-status Report
  • [Remarks] Publications and preprints

    • URL

      https://www.kurims.kyoto-u.ac.jp/~nkajino/publications.html

    • Related Report
      2023 Annual Research Report 2022 Research-status Report 2020 Research-status Report
  • [Remarks] 梶野 直孝|京都大学数理解析研究所

    • URL

      https://www.kurims.kyoto-u.ac.jp/ja/list/kajino.html

    • Related Report
      2023 Annual Research Report 2022 Research-status Report
  • [Remarks] Publications and preprints

    • URL

      https://www.kurims.kyoto-u.ac.jp/~nkajino/

    • Related Report
      2021 Research-status Report
  • [Remarks] 京都大学数理解析研究所・メンバー・梶野 直孝

    • URL

      https://www.kurims.kyoto-u.ac.jp/ja/list/kajino.html

    • Related Report
      2021 Research-status Report 2020 Research-status Report
  • [Remarks] Publications and preprints

    • URL

      http://www.math.kobe-u.ac.jp/HOME/nkajino/publications.html

    • Related Report
      2019 Research-status Report 2018 Research-status Report
  • [Remarks] 神戸大学理学部数学科・各教員の研究活動・梶野 直孝

    • URL

      http://www.math.kobe-u.ac.jp/home-j/index7-2-27.html

    • Related Report
      2019 Research-status Report 2018 Research-status Report
  • [Funded Workshop] Analysis and geometry of fractals and metric spaces: Recent developments and future prospects2023

    • Related Report
      2022 Research-status Report

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Published: 2018-07-25   Modified: 2025-01-30  

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