Some Chain Recurrent Properties Of Continuous Self-Maps On Compact Metric Space

All along, the study of the chain recurrent properties of some contin-uous mapping on compact metric space play an important part in topolo-gy dynamical systems. In this article, the defines of the Strong chain equivalent set and the Like- tent map on compact metric space are given, and we obtained some conclusions.In chapter 1, we introduce the origin and the development of the dynamical system and the background of the Like-tent map.In chapter 2, we introduce the necessary knowledge of the basic topology and classical discrete dynamical system, which closely related in this paper.In chapter 3, we present some properties of strong chain equivalent sets and strong chain recurrent set with the continuous maps of a compact metric space into itself. Further more, we show the connection between chain recurrent point and strong chain recurrent point. Finally, we give an example to show this result.In chapter 4, we research the periodicity and chain recurrent properries of Like-tent maps on the interval,we get two conclusions:1.When 0≤λ≤1/2,CR(fλ)=SCR(fλ)={0}u[λ,1].When 1/2<λ≤1, CR(fλ)=SCR(fλ)=Fix(fλ)={0,λ-2/2λ-3}.2.When 0≤λ≤1/4,fλ has periodic points of arbitrary positive integer.3.There exists x∈[0,1]such that CE(x,fλ)≠SCE(x,fλ).