Some Studies Of Transitive Properties In Topological Dynamical Systems

Given a dynamical system, to study its complexity is a central problem in the study of dynamical systems. The research of dynamical systems is varied, but the study of the asymptotic properties of orbits as well as its topological structure is essential. And the asymptotic properties of orbits or its topological structure is closely related to the recurrent time of points or open sets. In this thesis, we mainly study the transitivity of a dynamical system. This thesis is divided into four chapters.In chapter one, we give a review about the historical background and some known achievements which are important for this thesis in topological dynamical systems.In chapter two, we mainly study some topologically mixing properties of a class of sub-shifts and get one necessary condition and two sufficient conditions for a sub-shift to be topologically mixing, and one sufficient condition for a sub-shift to be topologically weak mixing.In chapter three, by the recurrent properties of uniform convergence functions and the properties of(quasi) weakly almost periodic points, we get some equivalent conditions the limit function of uniform convergence functions to be of some transitive property on dynamical systems.In chapter four, we mainly use recurrent time to study the transitivity of a sub-dynamical system, and by using the concept of Furstenberg family, we get some sufficient conditions for a dynamical system to be topologically weak mixing.