分类问题是拓扑动力系统研究的一个重要方向。关于拓扑动力系统中极小系统的分类已经有非常丰富的结果。本项目旨在研究传递系统的分类及其相关课题。2011年申请人提出传递系统的一个新的分类方式：利用传递点进入开集的时间集对传递系统进行分类。我们准备用这个新的分类方式来刻画弱混合和强混合属性。2010年Moothathu引入多重传递，Δ-传递和多重极小的概念。2012年Kwietniak 和Oprocha 他们通过举例说明多重传递属性和弱混合性质是两个不同的性质。他们提出下面的公开问题：能否用适当的方式来刻画多重传递和多重极小属性？我们准备肯定的回答这个问题。通过仔细分析这两类系统的传递点进入开集的时间集，我们考虑用合适的Furstenberg族来刻画这些时间集，从而得到多重传递和多重极小属性的刻画。此外，我们还研究拓扑动力系统与组合数论的联系，将传递系统的分类结果应用到组合数论中。

The study of classification of topological dynamical systems is one of the most important research areas of dynamics. There are plenty of results in the classification of minimal systems. This project is devoted to study the classification of topological transitive systems and its related topics. In 2011, we introduced a new way to classify transitive systems by the entering time sets of a transitive point into open subsets. We plan to use this new way to get a new characterization of weak mixing and strong mixing systems. In 2010, Moothathu introduced the concepts of multi-transitivity, Δ-transitivity and multi-minimality. In 2012, Kwietniak and Oprocha showed that in general there is no connection between multi-transitivity and weak mixing by constructing examples. They also proposed an open problem: is there any non-trivial characterization of multi-transitive systems and multi-minimal systems? We are going to give a positive answer to this problem. By studying the entering time sets of a transitive point into open subsets of those two classes of system, we plan to use proper Furstenberg families to describe the entering time sets, and then give some characterizations of multi-transitive systems and multi-minimal systems. In addition, we will also study the connection between topological dynamics and combinatorial number theory, and use the results of the classification of transitive systems to study combinatorial number theory.