本项目研究Moran集的分形结构及其应用，它们在理论与应用上有重要意义。Moran集是一类非常典型，非常广泛的分形集类，它包括所有熟知的分形集。本项目第一个课题研究Moran集的一些基本问题，包括一般Moran集（特别是压缩比下确界为零的情形）的维数，纲性，高维Moran集，共形Moran集，图递归Moran集，随机Moran集等，Moran集类的整体性质，子类的构造，不同拓扑的比较。并将它们用于其它分形结构。第二个问题研究Moran集的双Lipschitz等价，这是分形几何与几何测度论的基本问题，目前仅对于自相似集有结果。第三个问题是Moran集在其它学科的应用，研究具准周期势的离散薛定谔算子的谱的结构，算子谱具有典型的不同层次不同类型的Moran结构，不但需要综合已知的典型技巧，还需要发展新的方法与技巧。通过研究这些基本基本问题，发展新的思想、方法和技巧，从而推动分形几何的发展。

The proposal studies the fractal stucture of Moran sets and some applications, which are important both in theory and practice. Moran set is a very tipical and large fractal class which contains all familiar fractal sets. The first aim of the proposal consists of studing some basic problems of Moran sets, containing the dimensions of general Moran sets, conformal Moran sets, graph-directed Moran sets, random Moran sets; global properties of Moran sets, construction of sub-class, comparison of different topology, which will be used to other fractals. The second problem is to study the bi-Lipschitz equivalence of Moran sets that is a basic question in fractal geometry and geometric measure theory,up to now only very few results for self-similar sets are known. The third problem is some applications in other fields, for example, structure of spectra of Schrodinger operator with quasi-periodic potentials which possesses different types of Moran structures, we will develop some new methods and techniques. By studing these besic problems, we will develop some new ideas, methods and techniques for fractals.