摘要
目的主要对动力系统的极小性进行研究,针对几种特殊的系统给出该系统为极小的等价描述.方法首先,利用定义证明一集合是极小的当且仅当每一点的轨道在该集合中稠密.其次,利用反证法结合空间的序列紧致性对序列紧致空间上的同胚映射的极小性进行讨论.最后,针对紧致度量空间上的等度连续同胚,利用空间的极小性和紧致性,得到以空间中某有限个点的有限长轨道为中心,以ε2为半径的开邻域构成的有限子覆盖,并利用f的等度连续性,由该子覆盖构造出以空间任一点的有限长轨道为中心的开邻域所作成的有限子覆盖,进而得到所要结论.结果序列紧致空间X上的同胚映射f是极小的当且仅当对于每一个非空开集U X,存在n∈N使得Unk=-nfk(U)=X;紧致度量空间(X,d)上的等度连续同胚f是极小的当且仅当对于任意ε>0,存在N=N(ε)∈N,使得对于每个x∈X,集合{x,f(x),…,fN(x)}在X中ε-稠.结论为进一步研究动力系统的极小性提供理论基础.
Objective Focusing on minimality of dynamical systems, equivalent descriptions of minimality of some special systems are given. Methods Firstly, by definition it is proved that the minimality of a set is equivalent to the condition that the orbit of every point is dense in that set. Secondly, by reduction to absurdity and the sequentially compactness of the space, the minimality of a homeomorphism on a sequentially compact space is discussed. Finally, as to an equieontinuous homeomorphism on a compact metric space, from the minimality and compactness of the space, a finite subcover is acquired, which is ε/2 formed by open neighborhoods centered at the finite-lengthed orbits of some finite points with radius . And by the equicontinuousness of f, a finite subcover which is comprised by open neighborhoods centered at the finite-lengthed orbits of an arbitrary point of the space is constructed from the above subcover. Furthermore, the conclusion is obtained. Results The homeomorphism f on a sequentially compact space X is minimal if and only if for every nonempty open set U lohtain in X , there exists n∈N such that Uk^n=-nf^k （U） = X i equicontinuous homeomorphism f on a compact metric space （X, d） is minimal if and only if for arbitrary ε〉0 , there exists N=N （ε） ∈N, such that for everyx ∈ X , the set { x,f（X）,...,f^N（x） } is ε- dense in X. Conclusion This paper provides some theoretical bases for further research on minimality of dynami- cal systems.
出处
《河北北方学院学报（自然科学版）》
2009年第3期1-3,共3页
Journal of Hebei North University:Natural Science Edition
基金
河北省自然科学基金资助项目(A2008000132)
关键词
极小性
极小集
紧致度量空间
开覆盖
等度连续同胚
minimality
minimal set
compact metric space
open cover
equicontinuous homeomorphism
