迭代函数系的分离条件及其应用

The fractal sets we shall study in this program are compact sets defined by iterated function systems; they are the main objects of fractal geometry. We shall mainly consider the following problems. The first aspect is the connectedness of self-affine tiles, general self-affine sets and self-similar sets. The second aspect is on the dimension problems of fractal sets generated by bi-lipchitzan iterated function systems. Including conditions implying the equality of Hausdorff dimension and box dimension and the computation of these dimensions. The third aspect is on the calculation of the Lebesgue's measure of a self-similar or self-affine set when it contains interior points. The fifth aspect is on stochastic processes defined on fractal sets. Here, we mainly consider Markoff chains defined on augmented trees for iterated function systems, the related Green's function, the Martin kernel and the Martin boundary of the Markoff chain. Since these problems are related to the separation conditions of an iterated function system, we shall also further characterize separation conditions of iterated function systems.

本项目研究的分形集是指由迭代函数系生成的非空紧集K，这是分形几何的主要研究对象之一。主要研究如下几个方面的问题。一是自仿Tile的连通性、一般自相似集和自仿集的连通性问题。二是双Lipchitz迭代函数系生成的分形集K的维数问题。包括Hausdorff维数和计盒维数相等的条件，维数的计算方法等。三是一般自相似集和自仿集的勒贝格测度的计算方法（有内点时）。四是分形集K上定义的随机过程。这里，我们主要关注由迭代函数系定义的加边树（augmented tree）上的马氏链以及相应的格林函数、Martin核和Martin边界。因为上述问题的研究都与迭代函数系的分离条件相关，我们也要进一步研究迭代函数系的分离条件的刻画。

我们主要研究了分形几何中的三个方面的问题，一是自仿测度的谱理论，二是分形集的几何性质。三是分形集上的马尔科夫链的性质。这些领域都是分形几何的重要方向，近年来受到国内外学者的广泛重视，出现了一大批重要的研究成果。. 自仿测度的谱理论方面，我们完成了5篇学术论文，其中两篇发表于Journal of Functional Analysis。一篇“ On the spectra of self-affine measures with three digits”已经被 Analysis Mathematica接收，等待发表。完成待发表的论文两篇，分别是“Tree structure of spectral self-affine measures”（已经投稿到Journal of Functional Analysis）和 “Uniformity of Spectral Self-Affine Measures”（已经投稿到Advances in Mathematics）。. 分形集的几何性质方面，完成了5篇学术论文，其中4篇论文已经发表，还有一篇已经完成准备投稿。. 三是分形集上的马尔科夫链的性质方面，发表了一篇学术论文。. 这些成果解决了一些公开问题，发现了一些新的规律或者推广了其他研究者的结果。也被其他许多研究者引用。