Hyperspace Topological Dynamical Systems, Topological Entropy And Dynamical Systems Of Varying Parameters

In this thesis, we mainly study two problems: (1) the relationships of topological entropy ent^*(f) between the original topological dynamical system and its hyperspace dynamical system, (2) the dynamical properties of variable-parametric dynamical system. The paper is organized as follows:In the first chapter, the origin, development and main contents of the topological dynamical system and the variable-parametric dynamical system are presented, and some concepts which characterize the dynamical properties and complexity of the topological dynamical system are introduced. Then the background, status and applications of topological entropy, transitivity and chaos theory are recomendated. Finally, we retrospect the background as well as the status of the hyperspace dynamical system.In the second chapter, we focus our attention on the ralationships of topological entropy ent^*(f)^[1] between the topological dynamical system and its induced hyperspace system. And we prove that the topological entropy of base map is not more than the hyperspace's. We also get that, under certain conditions, if the base map has positive topological entropy then the corresponding hyperspace map has infinite topological entropy. These results coincide respectively with the conclusions in paper [2] and [3], in which the author investigate the relationships of topological entropy defined by Adler between the original topological dynamical system and its hyperspace dynamical system in compact metric space. However, the topological entropy ent^*(f) does not require compactness and measurement, therefore the results presented in this chapter make it possible to predict the complexity of the hyperspace dynamical system through the complexity of the original dynamical system in non-compact and non-metric space. In addition, we obtain some new properties of topological entropy ent^*(f): (1) if one topological dynamical system conjugate with another, then the topological entropy of the factor is not more than the topological entropy of the expansion; (2) ent^*(f^k) = k·ent^*(f); (3) ent^*(f^×k) = ent^*(f^*k). These properties are consistent with Adler topological entropy in compact space and Bowen topological entropy in metric space.In the last chapter, on the basis of theories in paper [4, 5], the notions of strong mixing, weak mixing, generator and expansion of the variable-parametric dynamical system are introduced, it turns out that in variable-parametric dynamical system strong mixing implies weak mixing and then implies transitivity; it is proved that if (X, F) and (Y, G) both are variable-parametric dynamical system, F conjugates with G , the members of F are communicate with each other and the members of G are also communicate with each other, what's more, they are both homeomorphism, then F is strong mixing (weak mixing, transitivity) implies G has the same properties; futhermore, we prove that F is strong mixing implies F Devaney chaos in the sense of modification in variable-parametric dynamical system and that F Devaney chaos in the sense of modification if and only if G Devaney chaos in the sense of modification when (X, F) semi-conjugate with (Y, G) and they both are communicate and homeomorphism; at last, we illustrate that F has generator if and only if it has weak generator, and we also prove that if F is expansion, then F has generator. These results concide with the theories presented in paper [6] and further extend the scope of the research on discrete system.