Study Of Some Points Set's Theories In Dynamical System

Some points of continuous self-mapping, such as periodic point, almost periodicpoint, eventually periodic point, recurrent point,ω- limiting point, non-wanderingpoint, chain recurrent point and so on, are one of the most contents of the topologicaldynamical system. For nearly thirty or forty years, plenty of scholars represented byXiong jincheng, Zhou zuoling, L.Block and R.Bowen have been interested in this areaand involved in it, and got plenty of important research results. However, with dynamicsystem's development towards high-dimensional and abstract space, how to generalizethe results of one-dimensional dynamical system to metric space or topological spacehas been an important problem. This paper deals with the problem and obtains someresults.In chapter 1, we briefly introduce the history of development about the topologicaldynamical system and the present research state, then,introduce the main content of thispaper.In chapter 2, we systematically the concepts and theorems involved in thetopological space, then recount the definition of the above-mentioned points andtheorems involved in the one-dimensional dynamical system.In chapter 3, the definition of the above-mentioned points is generalized in themetric space or topological space, some properties and inclusion relations of theabove-mentioned point sets are discussed, and some results are obtained. (1)The set ofperiodic point P(f) and the set of chain recurrent point P(f) are iterative, i.e.P(f) = P(f~n) , CR(f~n) = CR(f) ; (2)The set of eventually periodic point EP(f)and the set of chain recurrent point are invariant subsets of f , i.e.f (EP(f)) - EP(f) , f (CR(f)) - CR(f) . The set of point x ,sω- limiting pointis invariant subsets of f , the set of non-wandering pointΩ( f) is invariant subsets off k . (3) The set of non-wandering pointΩ(f) and the set of chain recurrentpoint P(f) are close sets. (4)The set of recurrent point, the set of eventually periodicpoint and the set of periodic point have the following relation, P(f) - EP(f) ,R(f)∩EP(f) = P(f) and so on.At last , the main efforts and results are summarized, and our view of perspectivesfor the future investigation is presented.