Aspects of Mathematics on Fractals

分形数学方面

• 批准号: 17340026
• 负责人: KIGAMI Jun
• 金额: $ 10.56万
• 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: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340026/
• 关键词: Fractal; heat kerenl; self-similar set; percolation; harmonic function; energy measure; tiling; 確率論; 力学系; 調和解析

We have obtained the following two results on relations between analysis and geometry on fractals. The first result concerns a heat kernel associated with a time change of a process associated with a self-similar Dirichlet form on a self-similar set. We show an sufficient and necessary condition on the existence of a distance under which the heat kernel satisfy the on-diagonal Li-Yau type sub-Gaussian estimate. The second result is on the measurable Riemannian Geometry on the Sierpinski gasket. We show that the geodesic distance coincides with the shortest path metric on the harrmonic Sierpinski gasket and establilsh Li-Yau type Gaussian estimate of the heat kernel under the geodesic metric.

关于分形的分析与几何的关系，我们得到了以下两个结果。第一个结果涉及与自相似集上自相似狄利克雷形式相关的过程的时间变化相关的热核。给出了热核满足对角线Li-Yau型亚高斯估计的距离存在的充要条件。第二个结果是关于西尔宾斯基垫片上的可测黎曼几何。我们证明了测地线距离与谐波Sierpinski垫片上的最短路径度量重合，并在测地线度量下建立了热核的Li-Yau型高斯估计。

项目成果

Random walk on the incipient infinite cluster on trees

• DOI: 10.1215/ijm/1258059469
• 发表时间: 2005-03
• 期刊: Illinois Journal of Mathematics
• 影响因子: 0.6
• 作者: M. Barlow;T. Kumagai

Measurable Riemannian geometry on the Sierpinski gasket

谢尔宾斯基垫片上可测量的黎曼几何形状

• 发表时间: 2007
• 作者: Kigami;Jun;木上 淳
• 通讯作者: 木上 淳

Measurable Riemannian geometry on the Sierpinski gasket : the Kusuoka measure and the Gaussian heat kernel estimate

Sierpinski 垫片上的可测量黎曼几何：Kusuoka 测量和高斯热核估计

• 发表时间: 2008
• 期刊: Math. Ann 340, no.4
• 作者: Kigami;Jun
• 通讯作者: Jun

Singular time changes of diffusions on Sierpinski carpets.

谢尔宾斯基地毯上扩散的奇异时间变化。

• 发表时间: 2006
• 期刊: Stochastic Process. Appl. 116
• 作者: Osada;Hirofumi,
• 通讯作者: Hirofumi,

Characterization of a uniform domain by the boundary Harnack principle. Harmonic analysis and its applications,

通过边界 Harnack 原理表征均匀域。

• 发表时间: 2006
• 期刊: Yokohama Publ., Yokohama
• 作者: Aikawa;Hiroaki
• 通讯作者: Hiroaki