本项目拟研究分形理论中的如下两个问题：分形集的Lipschitz等价问题和分形集的最佳参数化问题。分形集的Lipschitz等价问题有着丰富的数学内涵，和许多数学分支有着密切的联系，比如数论,有限自动机，概率论和动力系统等。而最佳参数化，作为一类具有良好性质的空间填充曲线，自1890年Peano里程碑式的工作以来一直是数学中的重要问题。我们拟研究下面几个问题：（1）非全不连通的分形的Lipschitz等价问题。在这个问题上，一方面我们寻找新的Lipschiz不变量，另一方面是利用有限自动机的理论，通过符号空间构造Lipschitz映射。（2）自仿集的Lipschitz等价。 我们希望利用自仿测度的重分形谱和加倍性作为Lipschitz不变量，来解决此问题。（3）图递归集、拟共形集和自仿集的最佳参数化问题。相比于自相似集，这些分形的最佳参数化在文献中很少触及。

This project is concerned with the Lipschitz classification and optimal parametrization of fractal sets. The Lipschitz classification is of great importance in its own right, regardness of its connection with problems from diverse fields such number theory, finite-state automaton, probability theory and dynamical systems. The optimal parametrization, as a class of space-filling curves with nice properties, is an important topic in mathematics since Peano’s monumental work in 1890. We shall study the following problems: (1) The Lipschitz equivalence of fractals which are not totally disconnected. On this problem, on one hand, we will try to find new Lipschitz invariant, and on the other hand, construct bi-Lipschitz maps via finite-state automaton. (2) The Lipschitz equivalence of self-affine sets. We will use multi-fractal fomalism and doubling properties of self-affine measures to attack this problem. (3) The optimal parametrization of graph-directed sets, self-conformal sets and self-affine sets. Comparing to the self-similar sets, the optimal parametrization of these fractals seems to be untrodden in literature.