Hypercontinuous Posets

The concepts of hypercontinuous posets and generalized completely continuous posets are introduced. It is proved that for a poset P the following three conditions are equivalent:(1) P is hypercontinuous;(2) the dual of P is generalized completely continuous;(3) the normal completion of P is a hypercontinuous lattice. In addition, the relational representation and the intrinsic characterization of hypercontinuous posets are obtained.     

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...     

树映射的链回归点与拓扑熵

Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced. It is proved that: (1) fhas zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f^5 has a division; (2) If f has zero topological entropy,then for any xECR(f)--P(f) the w-limit set of x is an infinite minimal set.     

ASYMPTOTIC BEHAVIOR OF THE DRIFT-DIFFUSION SEMICONDUCTOR EQUATIONS

This paper is devoted to the long time behavior for the Drift-diffusion semi-conductor equations. It is proved that the dynamical system has a compact, connected and maximal attractor when the mobilities are constants and generation-recombination term is the Auger model; as well as the semigroup S(t) defined by the solutions map is differential. Moreover the upper bound of Hausdorff dimension for the attractor is given.     

Likely Limit Sets of Feigenbaum''''s Maps and Their Hausdorf f Dimensions

A continuous map from a closed interval into itself is called a Feigenbaum's map if it is a solution of the functional equation f2(λx)=λf(x).In this paper, the likely limit sets of a type of Feigenbaum's maps are studied and their Hausdorff dimensions are estimated.As an application, we prove that for any 0相似文献   

On the extremal sets of extremal quasiconformal mappings

The properties of the extremal sets of extremal quasiconformal mappings are discussed. It is proved that if an extremal Beltrami coefficient μ(z) is not uniquely extremal, then there exists an extremal Beltrami coefficient v(z) in its equivalent class and a compact subset E Δ with positive measure such that the essential upper bound of v(z) on E is less than the norm of [μ].     

广义Fourier变换及其应用

It is proved that for each 2n×2n symplectic matrix S, there exists con-tinuous linear map F_s: S′(Rⁿ)→S′(Rⁿ), unique up to a constant factor, such that(?) F_s is called the generalized Fourier transformation. Some properties and applica-tions of F_s are obtained. In especial , the lower and upper boundedness of F_s in H^m(Rⁿ) is proved and a new proof of L. Hormander Theorem is given.     

Topological Anosov Maps of Non-compact Metric Spaces

Let X be a metric space. We say that a continuous surjection f: X→X is a topological Anosov map (abbrev. TA-map) if f is expansive and has pseudo-orbit tracing property with respect to some compatible metric for X. This paper studies the properties of TA-maps of non-compact metric spaces and gives some conditions for the map to be topologically mixing.     

The Entropy and Amount of Information of Fuzzy Events

In this paper,the entropy of random experiments with results of fuzzy events and the mutualinformation between two experiments of such kind are defined,therefore,the Shannon entropy andinformation are extended to the fuzzy situations.The principal properties of entropy and informationof fuzzy events are discussed.Among other things,we find that the entropy of a random experimentwith fuzzy outcomes is the sum of random entropy and fuzzy entropy.The connection betweenShannon entropy and De Luca Entropy is established.     

THE TOPOLOGICAL AND DYNAMICAL PROPERTIES OF NONORIENTABLE SURFACES

Four topological and dynamical properties of nonorientable surfaces are proved.Thefirst is that for every continuous flow defined on any nonorientable closed surface,thereexist periodic or singular closed orbits.In the case of the projective plane,it confirms aconjecture of professor Ye Yian-qian in his lecture notes“dynamical systems on surfaces”.Secondly,the author gives an exact upper bound of the number of closed curves onnonorientable surfaces,which do not intersect each other and the complement of their sumis still connected.The third is concerned with the upper and lower bound of the number ofthe periodic or singular closed orbits with certain properties.The last is related to theconnectedness of the complement of a lifting curve on two-fold covering space.The firstproperty may be considered as a generalization of Kneser theorem from Klein bottle togeneral nonorientable surfaces and the second as a generalization of[4]Theorem 9.3.6from orientable surfaces to nonorientable surfaces.     

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．     

树映射有异状点的一个充要条件

讨论了树上连续自映射的拓扑熵与非稳定流形之间的关系． 证明了：树上连续自映射有异状点的充要条件是其拓扑熵大于零． 因而推广了区间上连续自映射的一个结果．     

树映射的分布混沌

设T是个有限树,f是T上的连续映射．证明了f是分布混沌的当且仅当它的拓扑熵是正数．一些已知结论得到了改进．     

Topological sequence entropy for maps of the interval

A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

     

平均熵

设Ｔ为紧度量空间Ｘ上的连续自映射,ｍ为Ｘ上的Ｂｏｒｅｌ概率测度,通过把测度（拓扑）摘局部化,引入了Ｔ关于ｍ的平均测度（拓扑）熵的概念,它们分别为相应ｍ－测度（拓扑）混沌吸引子熵的加权平均,从而Ｔ关于ｍ的平均测度（拓扑）熵大于零当且仅当Ｔ有ｍ－测度（拓扑）混沌吸引子．证明了线段Ｉ上关于Ｌｅｂｅｓｇｕｅ测度平均拓扑熵大于Ｃ与等于零的连续自映射都在Ｃ０（Ｉ,Ｉ）中稠密．     

Variational principles and mixed multifractal spectra

We establish a ``conditional' variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.

Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the ``mixed' multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the ``non-mixed' multifractal spectra.

     

华沙圈及其推广的一些拓扑与动力性质

研究华沙圈及其推广上连续自映射的一些拓扑与动力性质,并通过对上半连续分解有关的某些动力性质的研究构作出华沙圈上的一个(在Devaney意义下的)混沌映射.     

上半连续集值优化解在图像逼近意义下的稳定性

给出上半连续集值映射优化问题在图像拓扑逼近意义下的本质弱有效解和本质有效解的概念.利用通有稳定性研究的usco方法, 证明了上半连续集值映射优化问题.在图像拓扑逼近意义下,弱有效解映射在定义域和映射同时扰动下是紧值上半连续的,从而是通有下半连续的,即在Baire纲意义下, 绝大多数上半连续集值映射优化问题, 在图像逼近意义下其弱有效解是稳定的,或者说是本质的. 证明了上半连续集值映射优化问题在图像逼近意义下有效解映射上半连续的一个充要条件,也即是有效解通有稳定的一个重要条件.     

非自治动力系统的原像熵

本文对紧致度量空间上的连续自映射序列应用生成集和分离集引入了点原像熵、原像分枝熵以及原像关系熵等几类原像熵的定义并进行了研究.主要结果是:(1) 证明了这些熵都是等度拓扑共轭不变量.(2)讨论了这些原像熵之间及它们与拓扑熵之间的关系,得到了联系这些熵的不等式.(3)证明了对正向可扩的连续自映射序列而言, 两类点原像熵相等,原像分枝熵与原像关系熵也相等.(4)证明了对(a).由闭Riemann 流形上的一个扩张映射经充分小的C1-扰动生成的自映射序列,以及(b).有限图上等度连续的自映射序列,有零原像分枝熵.     

树映射的单侧γ-极限点集与拓扑熵

本文讨论了树映射的单侧γ-极限点集与吸引中心的关系，得到了树映射具有正拓扑熵的几个等价条件．此外，还得到了树映射是强非混沌以及逐片单调树映射的拓扑熵为零的几个等价条件．