Spanning Set And Separating Set, The Bowen-Topological Entropy In The Product System

The topological entropy is one of the most important concept in the theory of the dynamical system of topology. It is a important conjugate invariance of topology. Its value can shows the chaos of the dynamical system. Therefore, in the dynamical system the study about topological entropy is very important study. In recent years, in the field of the dynamical system, one has done many researches and obtained a series of remarkable results. Our basic objection in this paper is to study the spanning set and separating set and the Bowen-topological entropy in the product system.In chapter one, we introduce simply the development about dynamical system and the background of this paper.In chapter two, we study mainly the characters of the spanning set and separating set and prove the following results: (1) f∈( X ,X)has certainly limited (n,ε)-spanning set; (2) Every (n+1,ε)-spanning set is the (n,ε)-spanning set; (3) Every (n,ε)-separating set is the (n+1,ε)-separating set ; (4)Every (n,ε)-separating set is a limited set. Finally, it discussed a character about the spanning set and separating set under the uniformly convergence.In chapter three, we discuss mainly the Bowen-topological entropy in the product system. We first discuss the relations of separating set and spanning set between system and product system. According to the results we further discuss the relations of the Bowen- topological entropy between system and product system and obtain that the Bowen-topological entropy of the product system is no bigger than the sum of the Bowen- topological entropy of each system, meanwhile, the Bowen-topological entropy of the product system is no smaller than the average value of the Bowen- topological entropy of all system.So we further obtain that the Bowen-topological entropy of power system to the power m is m times as large as the Bowen- topological entropy of system. In chapter four, first we summarize the results in this thesis, then we analyze the questions that exist in this studying and show the direction of studying that we will do in future.