Influence Of Space Properties On Dynamics Properties

There arc many fundamental theorems in topological dynamics concerning period, almost periodic point, recurrent point, ω - limit point, orbit closure, the relationships betwwen f and f~p and transitive map etc. It is well known that these theorems are true for compact metric spaces (sometimes also true for non-compact metric spaces). But for some dynamics properties that hold in all compact metric spaces, people usually take great pains to make it clear whether these properties also hold for much more general topological spaces. In this paper, we review these basic properties one by one, giving proofs to the properties that can be generalized to much more general or the most general topological spaces. But the methods are different from those familiar proofs available for compact metric spaces or metric spaces. While we present counterexamples for those properties that can't be extended to the most general or more general topological spaces.Besides, we discuss the mutual relationships among the four different definitions of the transitive map and give the equivalent condition of the four definitions in what spaces.