Devaney, Chaos Equivalent Portray Normed Space Is A Continuous Map Of Recurrent Points

As a new science,Chaology is thought usually begins in the paper:"Period three implies chaos"by Li Tianyan and James A.Yorke published in《Amer. Math. Monthly》in 1975, because"chaos"is first used as the scientific terminology in this article. Since the definition of Li-Yorke chaos is highly abstract and lacks intuitive,Therefore, Devaney has given an intuitive stronger Devaney definition in 1986. In recent 30 years a great advancement has been obtained in research of Chaos Theory and a number of research results is applied in many field of Natural Sciences and Social Sciences.Chaos mathematics foundation is very weak until now . So, it is currently a hot topits in chaotic mathematics basic research to seeks for equivalent characterizations between a various kind of chaotic definitions and the relations between them. One of this article aim is to discuss this question. First, The definition of Devaney chaos is generalized from a metric space to a topology space. Two groups of equivalent characterizations of Devaney chaos on a topology space are proved. As the corollary of the above results, the following is obtained: a continuous self mapping f :Xâ†'Xis chaotic in the sense of Li-Yorke if any two non-empty open subsets share a periodic orbit of f , where X is an interval or a compact metric space. Finally, two examples are revealed to illustrate the validity in the applications of the above results.Next, this article another studies the recurrent points of a continuous mapping from a normed space X to oneself ,obtained the following three results: (1) Let f be a surjective and injective continuous self-mapping from a sequentially compact normed space X to oneself, if x is a recurrent point of f , then there is some recurrent points x₀ of f such that fⁿ( x₀)= x for every positive integer n ; (2) If f is a continuous mapping from sequentially compact normed space to oneself, then the set of all recurrent points is a strong invariant set of f ; (3) If f be a continuous mapping from a loclly connected space to oneself, then every recurrent point x of f is an almost periodic point or an accumalation point of the set of all almost periodic points. A series of results obtained in this paper, to a certain extent, is a enrichmet and development of Chaotic Theory and Recurrent Points Theory in Topological Dynamic System and Applications of Topology are showed in our research .