动力系统与分形几何是两个独立但有密切关系的学科。申请者多年来形成动力系统-遍历理论-分形几何的研究方向，作出一系列好成果，并于近四年内完成两部专著，总结了过去的成果,提出一系列新问题。在动力系统中，我们曾提出测度中心和弱（拟弱）几乎周期点的概念，发现系统的不变测度和混沌均有三个层次，而遍历测度具有特殊的重要性，使动力系统研究得以深入一步。在分形几何中我们推翻了国外学者提出的两个猜测，使一个被长期尘封的重要概念- - -上凸密度，得以复苏，并形成满足开集条件的自相似集的Hausdorff测度理论研究和计算了一个新的理论体系（包括我们提出的新的研究领域- - -自相似集的结构）。我们的两项研究均属原创性的，而非跟在国外学者后面跑，因而我们提出的问题也带根本性，重要性经得起时间考验，影响深远，引起国内外学者竟相研究。我们提出的问题有些已解决，有些尚待解决。本课题目标是在两个领域深入研究我们自己提出的问题

Topological dynamical systems and fractal geometry are two indenpendent but closely related subjects in mathematics. The applicant has been working on dynamical system - ergodic theory - fractal geometry for many years, has obtained lots of significant results, and in the past 4 years we completed two monographs in these areas, which contain not only the newer developments and but also present many new problems. In the research of dynamical systems, we have introduced the notions of measure center and weakly and quasi weaklya lmost periodic point, to discover and describe that there are 3 different levels on invariant measures also on chaos，Especially , ergodic measure possesses important meaning. This is a further progress in the research of dynamics. In fractal geometry, we have given negative answers to two conjectures posed by a foreign scholar, such that the important notion- - upper convex density (which is neglected for a long time)- - now is widely concerned again. With this tool, it is formed the new system that is the research and computation of Hausdorff measure on self-similar sets with open set condition (including the new research area- - the structure of self-similar sets). The two research topics are originally presented, not just following foreign scholars, and the problems we presented are of fundamental importance, attracting the attention of domestic and international scholars. Some of the problems we presented have been solved, some remain unsolved. The target of this project is to study further our own topics in the areas.