Weakly Mixing Subset Of Topological Dynamical Systems

As is well-known, topological dynamical systems mainly investigate the limit behavior of the dynamical behavior as time goes to infinity. Among them, the study of chaos plays a very important role. And we know that the investigation of chaotic properties is crucial, because it not only is one of the important subjects, but also lays it the foundation on making further understanding chaos. In this thesis, we mainly study the dynamical properties of weakly mixing set which are chaotic properties and complexity of partial topological weakly mixing. The paper is organized as follows:In chapter 1, the origin, status and applications of chaos theory are de-scribed. Then the origin and the vital role in the chaos theory of sensitive dependence on initial conditions are introduced. Finally, the two basic chaotic properties—overall chaotic properties and partial chaotic properties are intro-duced. The introduction, background and status of weakly mixing sets are de-scribed.In chapter 2, it is narrated as the basis of our discussion, we give the primary notions of weakly mixing set and so on.In chapter 3, firstly, we give three equivalent conditions of a weakly mixing subset. Secondly, by using one of equivalent conditions of a weakly mixing subset, we investigate the dynamical behavior on limit of weakly mixing set, and under special conditions, we prove that the limit of a sequence of weakly mixing subsets is still a weakly mixing subset. Finally, we get a new property of weakly mixing set, it is an uncountable subset.