动力系统的很多问题都是跟分形有关的，尤其是由动力系统产生的分形集， 这些分形集往往需要分形几何中的工具和技巧 。 类似地， 动力系统中的很多技巧也都是可以移植到分形几何中，尤其是研究自相似集。任何一个自相似集都可以看作是一个拓扑动力系统， 因此我们通过动力系统来研究自相似集。本课题利用动力系统中一些想法对一些重叠的自相似集（或者一些开系统产生的分形集）做个马尔可夫分割， 这样就把原来有重叠的自相似集（或者一些开系统产生的分形集）转变成满足开集条件或者强分离条件的有向图自相似集合。有了这个转换， 一些问题就会变的容易一些， 比如，下面的两个问题： 某些自相似集中具有多个码的点集的分析， 由某些开系统产生的分形集的分析。 总之，通过本项目的研究我们希望能在研究重叠自相似集和一些分形集上的方法上有所创新。

Many problems in dynamical systems are related to fractal geometry, in particularly some fractal sets generated by some dynamical systems. Usually one needs some techniques in fractal geometry to study these sets. Similarly, many ideas in dynamical systems can be utilised in fractal geometry, for instance, to investigate the self-similar sets. Any self-similar set can be viewed as a topological dynamical system, therefore, we may study the self-similar sets in terms of the dynamical systems. In this project we make use of some ideas in dynamical systems to give a Markov partition for some self-similar sets or some fractal sets generated by some open dynamical systems. Subsequently we transform the self-similar set or the survivor set generated by open dynamics into a graph-directed self-similar set with the open set condition or the strong separation condition. Once we have this transformation, some problems could be easier, for example, the following two problems, analysing the set of points with multiple codings for some self-similar sets, investigating some fractal sets generated by the open dynamical systems. In brief, we hope that some new techniques can be found by investigating overlapping self-similar sets and some fractal sets.