Mathematical Fundation of Fractals

这个项目的目的是从分析、概率、遍历理论、动力系统和应用数学等不同的数学观点来研究分形。根据这个项目的目的，我们举行了两次会议。第一次是在项目第一年举行的。我们讨论了主要问题以及我们应该如何处理这些问题。在项目的最后一年举行的第二个项目是收集我们在这个项目中取得的所有成果。以下是从这个项目中挑选出来的结果。Kigami证明了在体积倍增条件下，热核的Li-Yau型上界估计等价于局部Nash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。伊藤研究了欧几里德空间的贝塔变换和相关的平铺。龟谷明确了Julia集与自相似集之间的关系。Hino已经证明了与Sierpinski垫片上的自相似Dirichlet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。

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.