Promotion Of Some Kind Of Generalized Periodic Points In Topological Spaces And Their Associated Properties

Generalized periodic point of continuous self-mapping, such as recurrent point, nonwandering point, chain recurrent point, is one of the most important researches of topological dynamical system. For almost 30 years, scholars at home and abroad are very interested in this area and have been actively involved in. They discussed the generalized periodic points on real segment, even metric space in depth, and got a lot of important research results. However, with modern dynamic system keeps the constant development to high-dimensional space and abstract space, some questions turn up: How to generalize the results of generalized periodic point on topological dynamical system to abstract topological space? Are there any important properties of generalized periodic point after the promotion?This paper deals with the above questions and obtains some results.Firstly, the concepts of periodic points, recurrent points and nonwandering points on real segment are generalized. In general topological space, some properties of periodic points, recurrent points and nonwandering points of continuous self-map are obtained; moreover, the correctness of these properties has been verified.Secondly, the concepts of unstable manifold on real segment are generalized. In general topological space, some properties of unstable manifold of continuous self-map are obtained; moreover, the correctness of these properties has been verified. At last, the paper describes the relationship among unstable manifold, periodic points,ω- limiting points, nonwandering points and homoclinic points.At last, the concepts of chain recurrent points andω- limiting points on real segment are generalized to metric space. Metric space is a special topological space. It is shown some properties of the chain recurrent points andω-limiting points in metric space. So, the corresponding results of closed interval have been improved.To a certain extent, these results enrich and promote the basic theory of periodic point in topological dynamical system. They lay the theoretical foundation of abstract development in dynamical systems theory and topology promotion in chaos mathematical theory.