Dynamical Properties Of Set-valued Discrete Systems

In this thesis, we investigate the inherent relations of dynamical properties between the dynamical system of compact metric space and its induced system: the set-valued discrete system along with its invariable subsystem.We retrospect the history as well as evolution of topological dynamics and chaos in section one. Furthermore , a number of basic notions of the dynamical system are introduced in this section which we will use in the following.In the second section, we focus our attention on the mixing properties, show that while the set-valued discrete system endowed with the topology which is induced by the metric of itself, some mixing properties such as specification property, strong mixing property, mild mixing property and weak mixing property etc. are equivalent between the basic space and the hyperspace . At the same time, a few of counterexamples are enumerated to indicate that some other properties haven't the equivalence between the two spaces, for example: transitivity , totally transitivity , Devaney chaos and so on. However, we present a new and weaker topology, proofing that the above properties are equivalent between the two spaces while the hyperspace endowed with the new topology.In the last section, we study the mixing properties and chaos of invariable subsystem of set-valued discrete system and its basic system, having the conclusion that most ,if not all, of the mixing properties of the induced system can be inherited to the basic one under some confined conditions. Moreover, we illuminate that using the idea of family to testify the theorems can largely simplified some proofs mentioned in section 2 .