Some Conditions Implying Continuity of Maps

In this note, we study continuity of maps, in topological spaces, in terms of cluster points of images of convergent sequences and nets, and obtain some analogues of two results of Fuller [1].     

Topological Stability and Entropy for Certain Set-valued Maps

In this paper,the dynamics(including shadowing property,expansiveness,topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view.It is shown that(1) if f is a hyperbolic endomorphism then for each ε> 0 there exists a C¹-neighborhood U of f such that the induced set-valued map F_f,U has the ε-shadowing property,and moreover,if f is an expanding endomorphism then the...     

关于映射同伦类的拓扑熵

设ｆ：Ｔ＾ｍ→Ｔ＾ｍ为ｍ维环面自映射，Ｎ＾∞（ｆ）是ｆ的渐近Ｎｉｅｌｓｅｎ数，本文应用Ｎｉｅｌｓｅｎ不动点理论，给出了ｌｏｇＮ＾∞（ｆ）是ｆ的同伦类的拓扑熵的最好下界的一个充要条件；并通过在齐性空间上引入等价度量，将此结论推广到了幂零流形自映射的情形。     

A Note on Topological Entropy of Maps

For a given collection of subsets of a compact metric space X which cover X,the entropies,hm(f)and hi(f),are introduced for a continuous map f of X to itseif. Also the concept of shadowing property with same end point (SPSEP for short) is introduced. The main results are,(1)H(f)≤hi(f)+hm(f)and,(2)h(f)=hm(f)whenever f has SPSEP,where h(f) is the topological entropy of f. Moreover,several corollaries are obtained.     

图映射拓扑序列熵的可交换性

主要研究图上连续自映射拓扑序列熵的可交换性,证明了对任意无界的正整数递增序列A=(ai)∞i=1和任意的连续图映射f,g都有hA(fog):hA(g of).解决了Balibrea F等人在相关文献中提出的一个猜想.     

圆周上单调映射的拓扑熵

本文研究了圆周上单调映射的拓扑熵，得到了圆周上连续单调映射ｆ的拓扑熵ｈ（ｆ）＝ｌｏｇ｜ｄｅｇ（ｆ）｜．     

线段映射的局部变差增长与局部拓扑熵

本文考虑闭区间上变差有界的连续映射f:I→I的局部变差增长γ(x,f)与局部拓扑熵h(x,f)．将证明γ(x,f)≥h(x,f)对所有x∈I成立,并且局部变差增长映射γf(x)=γ(x,f)与局部拓扑熵映射sf(x)=h(x,f)都是上半连续的,得到一个变分原理:局部变差增长γ(x,f)与局部拓扑熵h(x,f)的上确界分别等于全局变差增长γ(f)=limn→∞1/nln Var(fn)与拓扑熵h(f)．当映射f:I→I拓扑传递时,与Brin 和Katok对局部(测度)熵的讨论类似,我们证明,至多除一个不动点外,局部变差增长γ(x,f)与局部拓扑熵h(x,f)在开区间I°内恒为常值．     

Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras

Let A be a Banach algebra, and let E be a Banach A-bimodule.A linear map S:AE is intertwining if the bilinear map is continuous, and a linear map D:AE is a derivation if ¹D=0,so that a derivation is an intertwining map. Derivations fromA to E are not necessarily continuous. The purpose of the present paper is to prove that the continuityof all intertwining maps from a Banach algebra A into each BanachA-bimodule follows from the fact that all derivations from Ainto each such bimodule are continuous; this resolves a questionleft open in [1, p. 36]. Indeed, we prove a somewhat strongerresult involving left- (or right-) intertwining maps.     

树映射的不稳定流形,非游荡集与拓扑熵

设f是个端点数为n的树T上的连续自映射.本文得到了f的单侧不稳定流形与拓扑熵的关系,并证明了:（1）如果x∈i=0∞fi（Ω（f））-P（f）,那么,x的轨道是无限的;（2）如果f有一组可循环的不动点,那么h（f）≥In2（n-1）.     

Continuity of Almost Harmonic Maps with the Perturbation Term in a Critical Space

The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class (Zygmund class) L^n/2 log^q L, n is the dimension with n ≥ 3. They prove that when q > n/2 the solution must be continuous and they can get continuity modulus estimates. As a byproduct of their method, they also study boundary continuity for the almost harmonic maps in high dimension.     

Entropy,Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers ε_n(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IR^m and S^m. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers ε_n(f) have the same asymptotics its ω(f, ε_n(X)) for f ε C(X). Finally, the results concerning functions f ε M(X) and f ε C(X) are transferred to operators with values in M(X) and C(X), respectively.     

关于S~1上扩张映射的拓扑熵的改进

对文献 [1 ]中圆周上扩张映射的拓扑熵的结论给予了改进 ,得到了结论 :设 f∈ Cr(S1 ,S1 )是扩张映射 ,则 ent(f) =log|deg(f) |.     

On the Continuity of Julia Sets and Hausdorff Dimension of N CP and Parabolic Maps

Denote by HD(J(f))the Hansdorff dimension of the Julia set J(f)of a rational function f.Our first result asserts that if f is an NCP map,and f_n→f horocyclically, preserving sub-critical relations,then f_n is an NCP map for all n(?)0 and J(f_n)→J(f)in the Hausdorff topology.We also prove that if f is a parabolic map and f_n is an NCP map for all n(?)0 such that f_n→f horocyclically,then J(f_n)→J(f)in the Hansdorff topology, and HD(J(f_n))→HD(J(f)).     

具有零拓扑熵的图映射的攀援集的测度

本文研究了具有零拓扑熵的图映射f的性质,证明了在任意有限f-不变的Borel 测度μ下,其攀援集的外μ-测度都是零.     

闭轨线上模映射的不动点类与拓扑熵下界估计

本文在自治系统dx/dt=f(x),f∈C(DRn,Rn)的闭轨线Γ上定义了模映射,并利用闭轨线Γ与单位圆周S1的同胚关系,给出了模映射的Reidemeister数、Nielsen数,以及模映射的拓扑熵下界估计.     

On the Set of Continuity of the Topological Entropy of Parameter-Dependent Mappings of the Interval

Functional Analysis and Its Applications - Families of continuous mappings of the interval continuously depending on a parameter are considered. Any $$G_delta$$ set dense in the parameter space is...