About Chaos And Semi-group Action In Topology Dynamical System

Since the 1880s, H.Poincar unveiled a prelude of dynamical systems theory; the research has a remarkable progress. Especially GD.Birkhoff and others abstracted the dynamical system defined by the classical differential equations to the topological dynamical system, and further improve this subject in theory, so that the power system become a branch of mathematics which has the most achievements in twentieth century and has achieved fruitful results. The result has been widely used in many fields; it becomes the mainstream of modern science-an important component in non-linear science.Chaos is a unique complex shape in dynamical system. There are a lot of methods to study the chaotic tools, especially the thought way of topology, it can avoid complicated calculations, it is a powerful tool for the study of chaos theory. In this paper, the author uses this method to study some characters of chaos, introduces the chaotic semi-group action and obtained some conclusions.First, this paper describes the development about dynamical system, and the background and main content of this paper. Secondly, the author mainly introduces some basic concepts of sets of points, and discusses the equivalence problem of sets of points that topological dynamical systems under the conditions of topologically transitive. Then, the author introduces the development of the definition of the chaotic and their relationships, focuses on the chaos and gives an equivalent condition. In addition, the author introduces the chaos semi-group action, studies some characters of chaos with semi-group action and the relationship between them, and promotes some properties under the action of chaos semi-group. Finally, the author summarizes this paper, analyzes the questions that exist in this studying and show the direction of studying in the future.