2004 Fiscal Year Final Research Report Summary
Mathematical Fundation of Fractals
| Project/Area Number |
14340034
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
KIGAMI Jun Kyoto Univ., Graduate School of Informatics, Professor, 情報学研究科, 教授 (90202035)
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| Co-Investigator(Kenkyū-buntansha) |
KUMAGAI Takashi Kyoto Univ., Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (90234509)
SHISHIKURA Mistuhiro Kyoto Univ., Graduate School of Science, Professor, 理学研究科, 教授 (70192606)
OSADA Hirofumi Kyushu Univ., Faculty of Mathematics, Professor, 数理学研究院, 教授 (20177207)
HATTORI Tetsuya Tohoku Univ., Graduate School of Science, Professor, 理学研究科, 教授 (10180902)
ITO Shunji Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (30055321)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Fractal / Dynamical System / Self-similar set / Tiling / Laplacian / heat kernel |
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| Research Abstract |
The purpose of this project is to study fractal from various mathematical viewpoints, for example, analysis, probability, ergode theory, dynamical systems and applied mathematics. We had two conferences in accordance with the purpose of this project. The first one held in the first year of the project. We discussed what was the main issues and how we should approach them. The second one held in in the last year of the project was to get together all the results we obtained in this project. The followings are the selection of results from this project. Kigami has shown that under the volume doubling condition, the upper Li-Yau type estimate of heat kernels is equivalent to the local Nash inequality and the escape time estimate. Kumagai along with Barlow and Bass has shown that the Li-Yau type heat kernel estimate is stable under a perturbation. Ito has studied beta-transform and the associated tiling of the Euclidean space. Kameya has made clear the relation between Julia sets and the self-similar sets. Hino has shown that the energy measure associated with the self-similar Dirichlet form on the Sierpinski gasket is mutually singular with any self-similar measure. Finally Kigami and Kameyama have obtained a relation between the topological property of a self-similar set and the asymptotic behavior of a diffusion process on it.
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