超空间与原空间动力系统之间的关系

设(X,d,f)为拓扑动力系统,其中X为局部紧可分的可度量化空间,d为紧型度量,f为完备映射,用2X表示由X的所有非空闭子集构成的集族,(2X,ρ,2f)为由(X,d,f)所诱导的赋予hit-or-miss拓扑的超空间动力系统.本文引入了余紧点传递和弱拓扑传递的定义.特别的,在X满足一定的条件时,给出了点传递,弱拓扑传递和余紧点传递之间的关系,并研究了(X,d,f)的余紧传递点,回复点和几乎周期点分别与(2X,ρ,2f)的传递点,回复点和几乎周期点之间的蕴含关系.这些结论丰富了赋予hit-or-miss拓扑的超空间的研究内容.     

Fell-拓扑的伪紧性（英文）

设X是拓扑空间，CL（X）表示X的所有非空闭子集的族，本文得到了下述结果：在CL（X）上的Fell-拓扑是伪肾的当且仅当X是feebly-紧或者非局部紧或者非σ－紧，由此得到了对于伪紧性不是闭遗传的两类新的拓扑空间。     

局部有限超拓扑的局部紧性

本文讨论了赋予局部有限拓扑的非空闲子集超空间的局部紧性.主要结果是:X正则,则其闭子集超空间局部紧当且仅当X可表示成一个紧空间与一个离散空间的拓扑和.     

关于非自治动力系统中的h-极小覆盖

设（X,d1,f1∞）与（Y ,d2,g1,∞）为两个非自治动力系统,h是从（X,d1,f．∞）到（Y,d2,g1∞）的拓扑半共轭．通过对自治动力系统中的h一极小覆盖的研究,本文得到了以下结论：1）对于任意的Y∈Y及X∈h-1（y）,orb（x,f1∞）被h映射为orb（y,g1∞）,w（x,f1∞）被h映射为w（y,g1∞）;2）在（X,d1,f1∞）中引入关于拓扑半共轭的h-极小覆盖的定义,证明了h一极小覆盖的存在性;3）对于任意的XEX和Y∈Y,在（w（z,f1∞）,f1∞。（x,f1,∞）与（w（y,g∞）,g1,∞（y,g1∞））均构成原系统的子系统的前提下,R（f1∞）被h映射为R（g1∞）．这些结论丰富了非自治动力系统的内容．     

完备度量空间中的混沌判定

本文研究完备度量空间上的离散动力系统的混沌标准,证明了如果完备度量空间X上的连续映射f具有正则非退化返回排斥子或连接不动点的正则非退化异宿环,则存在f的不变闭子集A,使得f限制在此不变闭子集上的子系统与两个符号的符号动力系统拓扑共轭,从而获得具有这类结构的连续映射f具有Devaney混沌、分布混沌、正拓扑熵及ω-混沌,此结果改进了已有的相关结果.     

hit-or-miss拓扑上的度量:直接扩充

设E是Hausdorff局部紧第二可数拓扑空间.用F表示由E的所有闭子集构成的超空间,其上赋予hit-or-miss拓扑.本文引入了E上的紧型度量和F上保距扩张的概念,建立了E上度量是紧型的充分必要条件,并且证明了E上任何一个紧型度量度可以直接扩充为F上的保距度量.     

诱导超空间连续流的拓扑传递性

(X,f)为紧拓扑空间X上的连续流,(K(X),f)是由(X,f)诱导的紧超空间K(X)上的连续流.研究了(X,f)和(K(X),f)拓扑传递性之间的关系.证明了如果(X,f)是拓扑传递的,那么(K(X),f)也一定是拓扑传递的,并且举例证明了其逆命题不成立.进一步证明了(K(X),f)拓扑传递的当且仅当(X,f)是弱混合的.     

超实数域^＊R的拓扑结构

研究了超实数域上两种有用的拓扑－Ｑ－拓扑和Ｓ－拓扑。证明了以下结果：空间（＊Ｒ，Ｑ）是完全不连通的；＾＊Ｒ的Ｑ－紧子集只有有限集；＾＊Ｒ中的每一个银河是（＊Ｒ，Ｓ）的一个连通分支；＾＊Ｒ中的每一个具有有限长度的区间（不必是闭的）都是Ｓ－紧的，同时也纠正了《Ｍａｔｈ．Ｊａｐｏｎｉｃａ》上一篇论文中关于＾＊Ｒ上的Ｑ－拓扑的性质的一些错误。     

按序列分布混沌与拓扑混合

本文讨论了按序列分布混沌与拓扑混合的关系,并证明了:若X为至少两点的可分局部紧致度量空间,连续映射f:X→X是拓扑混合的,则对于任一正整数递增序列{mi},存在X的c-稠密Fσ子集D是f按{mi}的某子序列的分布混沌集.     

MV-代数的素滤子MV-拓扑空间

在MV-方体[0,1]X的子集Ω上引进MV-拓扑结构,并套论MV-拓扑空间的紧性、Hausdorff分离性等拓扑性质.细致地讨论MV-代数的素滤子集上的MV-拓扑空间(M,ΩM),证明素滤子MV-拓扑空间是紧Hausdorff MV-空间,并且它还是良紧空间.作为应用,证明一个σ-完备格M是MV-代数当且仅当M同构于某个Stone MV-空间的MV-开闭集格.     

Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems

The concepts of collective sensitivity and compact-type collective sensitivity are introduced as stronger conditions than the traditional sensitivity for dynamical systems and Hausdorff locally compact second countable (HLCSC) dynamical systems, respectively. It is proved that sensitivity of the induced hyperspace system defined on the space of non-empty compact subsets or non-empty finite subsets (Vietoris topology) is equivalent to the collective sensitivity of the original system; sensitivity of the induced hyperspace system defined on the space of all non-empty closed subsets (hit-or-miss topology) is equivalent to the compact-type collective sensitivity of the original HLCSC system. Moreover, relations between these two concepts and other dynamics concepts that describe chaos are investigated.     

Fell—拓扑的某些性质

刻画了Fell-拓扑的某些性质以及Fell-拓扑和拓扑收敛的关系。     

Proximal Hypertopologies Revisited

It is the aim of this paper to prove that for an arbitrary metric space (X,d) and a set of nonempty closed subsets of X which contains all singletons and which is closed under enlargements, we can construct a canonical approach distance on the hyperspace CL(X), having the -proximal topology (resp. the Hausdorff metric) as its topological (resp. p-metric) coreflection. We investigate some properties like, e.g., compactness and completeness of the introduced approach structures. In this way we obtain results which generalize their classical counterparts for proximal hit-and-miss hypertopologies. We also give a characterization of the completion of the introduced approach spaces.     

关于连续变换的遍历性质的一些注记

设(X,d)是一个紧的距离空间,T是(X,d)上的连续变换.利用平均遍历定理证明了:对任意的x∈X,1/n sum from i=0 to n-1 f(T~i x)在C(X)上收敛.该结果是连续变换的Birkhoff型个别遍历定理的推广.由此结果研究了T的其它遍历性质,特别,不依赖深刻的Choquet积分表示定理,给出了遍历分解定理的一个较为简单而直接的证明.     

Decomposition Properties of Hyperspace Topologies

Let X be a T ₁ topological space and a nonempty family of closed subsets of X. We study the hit-and-miss hyperspace topology generated by in terms of its upper and lower parts, focusing on first and second countability and quasi-uniformization. We also obtain some new results on the Vietoris and Fell topologies.     

Topologies of the Hyperspace which Are Induced by Extension Spaces

The hyperspace of all nonvoid closed subsets of a topological space will be topologized by means of different methods; each of them generalizes known definitions of special hyperspace topologies. One method uses the VIETORIS topology in the hyperspaces of certain extension spaces. Another one uses certain systems of closed subsets and leads to just the same class of hyperspace topologies. In natural order VIETORIS topology is the supremum of this class, and for locally compact spaces FLACHSMEYER topology is the infimum of this class.