加倍测度在分形几何中的研究

The project plans to study the properties of doubling measures from the viewpoint of fractal geometry. We shall study the following two related problem: (1) discussing the singularity of doubling measure due to the geometric properties of the doubling space. (2) studying the pointwise dimension and the local structure of doubling measures, furthermore, discussing that for what doubling space, there is a doubling measure supported on it with nice property and the dimension of the doubling measure equals to the dimension of the doubling space..The study of the project relates to the basic theory of fractal geometry and doubling measures, the result of its is very helpful to understand the geometric structure of fractals on doubling space and develop the theory of doubling measures. It is important in academic study.

本项目计划从分形几何的角度来研究加倍测度的性质。主要研究下面两个相关问题：（1）讨论由加倍空间的几何性质导致的加倍测度的奇异性。（2）研究加倍测度的点态维数和局部结构，讨论什么样的加倍空间能支撑性质较好的加倍测度，使得这个加倍测度的维数就等于空间自身的维数。.本项目的研究涉及分形几何的基本理论与加倍测度理论的交叉，其研究结果有助于理解加倍度量空间上分形集的结构、进一步丰富和发展加倍测度理论，具有很重要的意义。

本项目从加倍测度的角度来研究分形几何，首先我们得到一类特殊的Moran型集是加倍测度肥（瘦）集的充要条件；其次在Lipschitz 等价条件下，我们研究了在某种意义下不是Ahlfors-David正则的齐性分形集被它们的尺度函数唯一决定。另外，本项目还研究了一类自相似集重分形分支的量纲。