The Study Of The Turbulence Of The Topological Dynamical System

Turbulence is an essential concept in researching topological dynamical sys-tem. Usually people use it as a condition when they study whether topological dynamical system contains symbolic dynamical system, contains chaos and so on. But what are the conditions ensuring that a topological dynamical system implies turbulence, especially, in the general metric space. There are few people to research it. In this paper the question which has been mentioned above is what we care. It is organized as follows:Chapter1introduces the reader to the background, the current develop-ment situation and the applications of dynamical system, topological dynamical system, chaos, symbolic dynamical system and turbulence. Then it gives the thinking of investigating problems. Finally, the purpose and the main achieve-ments are presented.In chapter2, firstly, we give the concepts of source fixed point and open convergence and find their example in symbolic dynamical system. Secondly, the basic properties about source fixed point and open convergence are established by using topological methods. Thirdly, we get the equivalent descriptions of them and point out that source fixed point and open convergence are preserved under topological conjugate.Chapter3and4, by using source fixed point and open convergence we ex-plore the condition about existence of turbulence in metric space. Finally, the sufficient and necessary conditions about topological dynamical system topolog-ically (semi-conjugate) conjugate to symbolic dynamical system is obtained, and it is used in symbolic dynamical system.