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.

本计画的目的是从分析、机率、遍历理论、动力系统及应用数学等不同的数学观点来研究分形。我们根据这个项目的目的举行了两次会议。第一次是在项目的第一年。我们讨论了什么是主要问题以及我们应该如何处理这些问题。第二次是在项目的最后一年举行的，目的是收集我们在这个项目中取得的所有成果。以下是该项目的成果选择。Kigami证明了在体积加倍条件下，热核的上Li-Yau型估计等价于局部Nash不等式和逃逸时间估计。Kumagai沿着Barlow和Bass证明了Li-Yau型热核估计在扰动下是稳定的。伊藤研究了β-变换和相关的欧氏空间的平铺。Kameya明确了Julia集与自相似集之间的关系。日野证明了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度与任何自相似测度都是相互奇异的。最后，Kigami和Kameyama得到了自相似集的拓扑性质与其上扩散过程的渐近行为之间的关系。

项目成果

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)

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。

Hambly, B.M., Kumagai, T.: "Diffusion processes on fractal fields : heat kernel estimates and large deviations"Probab. Theory Related Fields. 127. 305-352 (2003)

Hambly，B.M.，Kumagai，T.：“分形场上的扩散过程：热核估计和大偏差”概率。

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。

Kigami, Jun: "Harmonic analysis for resistance forms"J.Funct.Anal.. 204-NO.2. 399-444 (2003)

Kigami, Jun：“阻力形式的谐波分析”J.Funct.Anal.. 204-NO.2。