Recurrent And Asymptotical Properties In Dynamical Systems

The aim of this paper is to discuss the recurrent and asymptotical properties in dynamical systems.In Chapterâ…¡, we discuss two important invariant sets, prolongation and prolongational limit set. Let X be a locally compact metric space, for a point x∈X, we get some properties of the higher positive prolongation D_α⁺ (x) and higher positive prolongational limit set J_α⁺ (x). In general, D_α⁺ ( D_β⁺(x))is not equal to D_α⁺ +β(x), but if z∈D_α⁺ (y)and y∈D_β⁺ (x) , then there is an ordinalηsuch that z∈D_η⁺ (x). For a set M ? X, we also discuss the positive prolongations D_α⁺ (M), D_u⁺ (M)and positive prolongational limit sets J_α⁺ (M), J_u⁺ (M), and get some relations among them. Even if M ? X is a compact set, J₁⁺ (M)=J u+(M) may not be true. We also obtain some results about the connectedness of the prolongation and prolongational limit sets. If M ? X is connected, D_α⁺ (M) is compact, then D_α⁺ (M) is connected. If M ? X is compact and connected , J_u⁺ (M) is compact , then J_u⁺ (M) is connected. At last, we give a theorem of stability. Let a dynamical system ( X ,R,Ï€) be positively Lyapunov stable at every point x∈X,and J₁⁺ (ω(x))=ω(x),ω(x )≠φ,then D₁⁺ (x) is orbit stable.Chapterâ…¢is devoted to the dynamical systems which are in general lack of recursiveness, and notably marked by the absence of Lagrange stable motions or Poisson stable points or non-wandering point. We give some equivalent conditions among them. Let X be a locally compact Hausdorff space, if the dynamical system ( X ,R,Ï€) is divergent, the orbit space X Cis a T1 space. If the dynamical system ( X ,R,Ï€)is dispersive, the orbit space X Cis a T₂ space. If the dynamical system ( X ,R,Ï€)is Lagrange unstable, and is positively Lyapunov stable on the set M = { x: J_α⁺ (x)≠φfor each ordinalα},then R =∪｛x∈J_α⁺ (x):αis an ordinal}=φ. In Chapterâ…£,we discuss some problems about minimal flows. Since there always exist minimal sets in a compact Hausdorff space , and the role that the minimal set plays in dynamical systems is like that the ergodicity does in measure-preserving systems. So it is one of the most important invariant sets in dynamical systems. There are two main aspects in the studies of...