Mathematical Fundation of Fractals

分形数学基础

• 批准号: 14340034
• 负责人: KIGAMI Jun
• 金额: $ 9.15万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340034/
• 关键词: Fractal; Dynamical System; Self-similar set; Tiling; Laplacian; heat kernel; 拡散過程; ジュリア集合

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.

该项目的目的是从各种数学角度研究分形，例如分析、概率、ergode 理论、动力系统和应用数学。我们根据这个项目的目的召开了两次会议。第一次是在该项目的第一年举行的。我们讨论了主要问题是什么以及我们应该如何解决这些问题。第二次是在项目的最后一年举行的，目的是汇总我们在该项目中获得的所有成果。以下是该项目的成果选择。 Kigami证明，在体积倍增条件下，热核的上Li-Yau型估计等价于局部纳什不等式和逃逸时间估计。 Kumagai 与 Barlow 和 Bass 一起证明了 Li-Yau 型热核估计在扰动下是稳定的。伊藤研究了 β 变换和欧几里得空间的相关平铺。 Kameya明确了Julia集和自相似集之间的关系。 Hino 已经证明，与谢尔宾斯基垫片上的自相似狄利克雷形式相关的能量测量与任何自相似测量都是互奇异的。最后Kigami和Kameyama得到了自相似集的拓扑性质和其上扩散过程的渐近行为之间的关系。

项目成果

Hambly, B.M., Kigami, I., Kumagai, T.: "Multifractal formalisms for thelocal spectral and walk dimensions"Math.Proc.Cambridge Philos.Soc.. 132-3. 555-571 (2002)

Hambly，B.M.，Kigami，I.，Kumagai，T.：“局部光谱和行走维度的多重分形形式”Math.Proc.Cambridge Philos.Soc. 132-3。

Osada, H.: "Harnack inequalities for exotic Brownian motions"Kyushu J.Math.. 56-2. 363-380 (2002)

Osada, H.：“奇异布朗运动的 Harnack 不等式”Kyushu J.Math.. 56-2。

Kameyama, Atsushi: "On Julia sets of postcritically finite branched coverings. I. Coding of Julia sets"J.Math.Soc.Japan. 55. 439-454 (2003)

Kameyama，Atsushi：“论 Julia 集的后临界有限分支覆盖。I. Julia 集的编码”J.Math.Soc.Japan。

Hino, Masanori, Ramrez, Jos A.: "Small-time Gaussian behavior of symmetric diffusion semigroups"Ann.Probab.. 31. 1254-1295 (2003)

Hino, Masanori, Ramrez, Jos A.：“对称扩散半群的小时高斯行为”Ann.Probab.. 31. 1254-1295 (2003)

Ito, Shunji, Fujii, Junko, Higashino, Hiroko, Yasutomi, Shin-ichi: J. Number Theory. 99. 255-283 (2003)

伊藤、俊二、藤井、顺子、东野、弘子、安富、新一：J. 数论。