关于一类自映射轨道的研究

1 概念及已有结果 设X为拓扑空间,f∈C0（X,X）,f0表示恒等映射,对任意自然数n,定义fn=fοfn-1. 称O（x,f）={fn（x）│n=0,1,2,… ;x∈X}为x的f轨道. 关于周期点、周期点集、周期、周期轨道,Sarkovskii序如通常定义,可参见[1].     

一类连续体上连续映射的周期点

设X是个阶有限的遗传可分解可链连续体, f:X→X是X上的连续自映射, On(x,f)={fi(x):0≤i≤n)是f的一个返回轨道, inf(On(x,f))相似文献   

改进的Tsirelson空间TM上的Wigner定理

设X和Y是赋范空间，如果映射f：X→Y满足{||f（x）+f（y）||，||f（x）-f（y）||}={||x+y||，||x-y||}（x，y∈X），则称f是一个相位等距算子.设g，f：X→Y是映射，若存在相位函数ε：X→{-1，1}，使得ε·f=g，则称g和f是相位等价的.本文将证明改进的Tsirelson空间T_M上的任意满相位等距算子均相位等价于一个线性等距算子.该结论同时也给出了改进的Tsirelson空间T_M上的Wigner型定理.     

真拟弱几乎周期点和拟正则点

T为紧致度量空间X上的连续映射,M（X）为X上所有Borel概率测度.设x∈X,记Mx（T）为概率测度序列{1n∑n 1i=0δTi（x）}在M（X）中的极限点的集合,其中δx表示支撑集是{x}的点测度.记W（T）和QW（T）分别为T的弱几乎周期点和拟弱几乎周期点集.本文证明,如果（X,T）非平凡且满足specifcation性质,则存在x,y∈QW（T）／W（T）（称为真拟弱几乎周期点）,分别满足μ∈Mx（T）,x∈Supp（μ）和ν∈My（T）,y∈/Supp（ν）,回答了周作领等提出的公开问题.Mx（T）在弱拓扑中是紧致连通集,所以,要么是单点集,要么是不可数集.如果x∈QW（T）／W（T）,则Mx（T）是不可数集.一个自然的问题是,怎么刻画M x（T）是单点集的点x（这时x称为拟正则点）.本文给出M x（T）是单点集的充要条件.     

集值映射空间在紧开拓扑下的N_0性质

本文讨论了点紧致的连续集值映射空间在赋予紧开拓扑下的某些拓扑性质,证明了:若X,Y为N_0空间,则X到Y上的点紧致的连续集值映射族依紧开拓扑是N_0空间,从而将Michael的结论推广到更大的映射空间类上.     

两个映射定理

本文证明了:(1)设f是正规,等紧(isocompact)空间X到空间Y上的闭映射,则f是紧覆盖映射;(2)设f是正规,等紧空间X到Fréchet空间Y上的闭映射,则存在闭子集X′(?)X使f|x′是X′到Y上的既约映射;分别改进了Michael、Lanev映射定理,并利用(1)得到“闭映射保持正规、k-半分层性”以改进Lutzer关于k-半分层空间的映射定理。     

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.     

序列覆盖的闭映射保持可度量性

设f:X→Y是连续的满映射. f称为序列覆盖映射,若{y})是Y中的收敛序列,则存在X中的收敛序列{xn},使得每一xn∈f-1(yn);f称为1序列覆盖映射,若对于每-y∈Y,存在x∈f-1(y),使得如果{yn}是Y中收敛于点y的序列,则有X中收敛于点x的序列{xn},使得每一xn∈f-1(yn).本文研究度量空间序列覆盖的闭映射之构造,否定地回答了Topology and its Applications上提出的一个问题.     

不等式约束的向量优化中值映射的余切上图导数

考虑如下的参数向量优化问题minK{f(w,x)|x∈X,g(w,x)∈C},这里f:W×X→Y是从赋范空间W和X的积到另一个赋范空间Y的Hadamard可微的单值映射,K Y是一个尖闭凸锥,C是Banach空间Z中的一个尖闭凸锥,g:W×X→Z是一个Fréchet可微的映射.借助目标函数的导数、约束映射的余切导数及拉格朗日映射给出了值映射的余切上图导数的两个表示.     

可数对一映射及相关问题

本文研究了■0-sn-度量空间与度量空间之间的关系.利用特殊映射,获得了在序列空间中下述命题等价:(1)空间X是■0-sn-度量空间;(2)存在从度量空间M到X可数对一、序列商、σ映射f;(3)存在从度量空间M到X可数对一、序列商、σ映射f使得对每一个x∈X,■f-1(x)是σ-紧.推广了参考文献[3,4]中的一些结果.     

GENERALIZATIONS OF KY FAN＇S THEOREM ON BEST APPROXIMATIONS

In 1969, Ky Fan[3] proved that for any continuous function f from a compact convex subset M of a normed linear space X into X, there exists x ∈ M such that f (x) - x = dist( f (x), M). Since then, there have appeared several generalizations, extensions and applications of this result. This paper also deals with some extensions and generalizations of this result when the underlying spaces are convex metric spaces.     

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.     

拓扑动力系统研究的图像方法(英文)

This paper considers the graph sequence graph （f^n ）={x×f^n ： x ∈X} of the iterates f^n of a continuous map f ： X →X on a compact metric space （X, d）. The main aim is to introduce serval limit sets by using different recurrent time sets, and discuss the relation between dynamics of the underlying compact system （X, f） and the structure of these limit sets, from a topological or fractal viewpoint.     

Non-wandering sets of the powers of dendrite maps

Let(X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f)and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x ∈Ω(f)-Ω(f~n) for some n ≥ 2,then x ∈ EP(f).(2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = ?0(the cardinal number of the set of positive integers).     

赋范空间的点态非方常数

引入点态非方常数的定义并给出其等价表达形式，同时给出点态非方常数在赋Luxemburg范数Orlicz序列空间和Orlicz函数空间的估计以及在1^p和L^p空间的计算值。     

Division and k-th root theorems for Q-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.     

某些局部凸空间上相对弱紧集，相对弱列紧集，相对弱可数紧集的关系

对可分离的局部凸空间（Ｘ，τ），本文建立了相应的局部凸空间（Ｑｘ，Ｔｘ），利用它证明了当（Ｘ，τ）满足某些条件时（赋范空间满足这些条件），ＥＸ相对弱紧Ｅ相对弱列紧Ｅ相对弱可数紧，从而推广了Ｅｂｅｒｌｅｉｎ等人的工作，证明了在空间Ｄ（Ｒ￣ｎ），（Ｒ￣ｎ）和Ｄ＿（ＬＰ），１＜ｐ＜∞上前述三种弱紧性等价．     

The continuity of the metric projection on a subspace of finite codimension in the space of continuous functions

The closed subspaces of finite codimension of the space C(X) of all continuous real-valued functions on a compact Hausdorff space X, for which the set of elements of best approximations of every function f C(X) is nonempty and compact, are characterized. It is shown that if the compact Hausdorff space X is infinite, then C(X) has no subspace of a finite Codimension n > 1 which has a nonempty set of elements of the best approximation for an arbitrary function f 6 (X) and which has an upper-semicontinuous metric projection.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 531–539, April, 1976.