Dynamics Of Induced Systems On Hyperspaces

Let (X,f) be a dynamical system. The hyperspace of X is a specified collection of non-empty closed subsets of X with the Vietoris topology, which is very important for connecting one-dimension manifold with high-dimension manifold. Our basic object in this paper is to study the relationships of some dynamical properties between the compact metric space and its hyperspace.In section 2, some dynamical properties related to transitivity, density of periodic points set , and Devaney's chaos among the dynamical system (X,f), (2X,2f) and (C(X), C(f)) are studied. Which partly answered the question brought by Heribertoin in [12]: individual chaos implies collective chaos? and conversely? Namely, (X,f) is Devaney's chaos does not implies that (2X,2f) or (C(X),C(f)) is Devaney's chaos. As an application, it is shown that Devaney's chaos are stronger than the existence of uncountable s-Scrambled sets.In Section 3, the relationships of some limit behavior, such as distality, proximity, expansivity, equicontitiuity, uniform rigidity, psuedo orbit tracing property, among (2X,2f), (C(X), C(f)), and (X,f) are studied. The equivalence of equicontinuity (uniform rigidity, respectively) among (2X, 2f), (C(X), C(f)), and (X,f) is proved.