【作者】 张更容；

【导师】 曾凡平；

【作者基本信息】 广西大学 ， 基础数学， 2004， 硕士

【摘要】 设(X，f)为一个动力系统。X的超空间是指X的所有非空闭子集构成的集合赋以Vietoris拓扑，它是一维流形和高维流形之间的一个重要的联系纽带。本文主要考虑紧致度量空间上的动力系统与其超空间上相应的诱导系统之间的动力学性质的内在联系。 第二节我们讨论了(2~X，2~f)、(C(X)，C(f))、(X，f)之间关于传递性、周期点稠密性、Devaney’s混沌的关系；这些研究部分回答了Heriberto在文献[12]中提出的问题：individual chaos implies collective chaos?and conversely?具体的说，指出了(X，f)是Devaney’s混沌并不蕴含(2~X，2~f)或(C(X)，C(f))是Devaney’s混沌的。作为应用，我们得出Devaney’s混沌性质是严格强于具有不可数的s-Scrambled集。 在第三节中，我们研究了Distal性质、迫近性质、可扩性、等度连续性、一致刚性、伪轨跟踪性质等极限行为在(2~X，2~f)、(C(X)，C(f))、(C，f)之间的内在联系。证明了等度连续性(一致刚性)在(2~X，2~f)、(C(X)，C(f))、(X，f)之间是等价的。更多还原

【Abstract】 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.更多还原