Set-valued Mappings And The Set Of Recurrent Points

Let ( X , d )be a compact metric space and f :X→X be a continuous map. Letκ( X) be the space of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d and the set-valued mapping (f|—) :κ( X )→κ( X)induced by f be the map defined by (f|—)( A) = {f (a ) |a∈A}. In this paper we mainly investigate the relationships between the set of recurrent points of ( X , f )and the set of recurrent points of (κ( X ), (f|—)). And we obtain a series of results. The paper consists of three chapters.In chapter 1, we make out the background of problem-researching, the recent development of the research in this field and what we have done in this paper. And we introduce the conceptions and fundamental knowledge. Moreover, we illustrate the importance of our work in theory and practice.In chapter 2, we discuss set-valued mappings and the periodic points set. In section 1 we obtain that if P ((f|—))is closed then so is P ( f )but the converse of it is not necessarily true. And we give an anti-example to illustrate the converse of it is not necessarily true. And we prove that ifφ_ω( X ) =κ( X) and P ( f ) is closed then P ((f|—)) is closed. In section 2, we point out that the converse of corollary3.2 (ii) in [18] will be true when P ( f )is closed and the conclusion is stronger than ever, i.e. P ((f|—)) =κ( X) can imply P ( f )= X. We give an anti-example to illustrate that P ( f )= Xcan't imply P ((f|—)) =κ( X). We obtain that if X is finite then P ( f )= X can imply P ((f|—)) =κ( X). In section 3, we prove that if P ((f|—))=φthen so is P ( f ) but the converse of it is not necessarily true. And we show that if (f|—) is minimal then so is f and we give an anti-example to illustrate the converse of it is not necessarily true. In section 4, we obtain that if X = I or T and f is topologically transitivity then P ((f|—)) is dense inκ( X). In section 5, we prove that if X = I and P ((f|—)) is closed then (f|—) is not Devaney's chaotic.In chapter 3, we study set-valued mappings and the set of limit points, the set of recurrent points, the set of almost periodic points. We prove that if F is theω- limit point of (f|—) then F contains aω-limit point of f . It is pointed out that for any F ofκ( X) if F contains aω-limit point of f then F is theω-limit point of (f|—) under the condition of W~e-topology. We prove that if W ( f ) is closed then so is W ((f|—)) under the circumstance of W~e-topology. And we give a sufficient condition of R ((f|—)) or AP ((f|—)) being closed.