| Project/Area Number |
18K18720
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | Kyoto University (2021-2023)
Kobe University (2018-2020) |
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Principal Investigator |
KAJINO Naotaka 京都大学, 数理解析研究所, 准教授 (90700352)
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| Project Period (FY) |
2018-06-29 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| 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)
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| 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 |
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| 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).
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| 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．
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