On The Research Of Some Problems In Topological Dynamical Systems

In this paper we generalize a class of sets L(x₁,x₂) in topological dynamical systems. We study the relative properties of these generalized sets, and then give a characterization of equicontinous systems. At the same time, we also give a deep discussion about the relationship between dynamical systems and their enveloping semigroups by means of the concepts of category and functor.In the first chapter, we introduce some necessary definitions and signs.In the second chapter, we give further discussions about a class of sets L(x₁,x₂) introduced in paper [4] and [5]. Firstly, we generalize its definitions and study the relative properties of these generalized sets. Then we give a characterization of equicontinous systems . Secondly, the relati- onships between L(x₁,x₂),L(x₁,x₂,x₃), L(x₁,x₂,x₃,x₄),…,L(x₁,…, x_n)are discussed and some examples are given.In the third chapter, we define a covariant functor F₁ from the category of dynamical system T to the category of enveloping semigroup E and a contravariant functor F₂ from the category T to the category E﹡by means of the concepts of category and functor. In addition, the questions are discussed whether the enveloping semigroup of product space in the category T is equal to the product of the enveloping semigroups in the category E and the enveloping semigroup of inverse limit space in the category T is equal to the inverse limit of the enveloping semigroups in the category E .