树映射的链等价集与拓扑熵

本文讨论了树映射f的链等价集的性质,得到了f具有零拓扑熵的几个等价条件,并证明了:如果 f的一个链等价集是个无限集,那么这个链等价集的任何孤立点都是f的非周期的终于周期点.     

树映射的链等价集与拓扑熵

本文讨论了树映射f的链等价集的性质,得到了f具有零拓扑熵的几个等价条件,并证明了如果f的一个链等价集是个无限集,那么这个链等价集的任何孤立点都是f的非周期的终于周期点.     

拓扑群作用下度量空间中链回归点集的研究

仿造度量空间中链回归点的定义,给出了拓扑群作用下度量空间中G-链回归点的概念,并将度量空间中链回归点的一些结论,推广到拓扑群作用下度量空间中,得到如下结果:1)同胚伪等价映射f的G-链回点集等于它的逆映射f~(-1)的G-链回归点集;2)伪等价映射f的G-链回点集和G-链等价集对G强不变;3)同胚等价映射f的G-链回点集f对强不变.4)等价映射f限制在它的G-链回归点集上形成的G-链回归点集就是等价映射f在度量G-空间X上形成的G-链回归点集.这些结果丰富了拓扑群作用下度量空间中G-链回归点的理论.     

遗传可分解可链连续体上的动力系统

证明遗传可分解可链连续体上,不含非2方幂周期轨道的连续映射限制到每个非周期回复点的ω极限集上拓扑半共轭于加法机器,得到Susline可链连续体上连续映射拓扑熵为0的五个充要条件.     

遗传可分解可链连续体上的动力系统

证明遗传可分解可链连续体上,不含非２方幂周期轨道的连续映射限制到每个非周期回复点的ω极限集上拓扑半共轭于加法机器,得到 Ｓｕｓｌｉｎｅ可链连续体上连续映射拓扑熵为 ０的五个充要条件．     

关于链同伦前伸的注记

首先,本文在链复形和链映射范畴d中证明了各种同伦扩张性质(HEP,WHEP,RWHEP与VWHEP)是等价的.这不同于拓扑空间和连续映射范畴Top.其次,在范踌d中也有Puppe序列,本文证明了约化奇异链函子保持Puppe序列的右正合性.最后,与范畴Top相类似,在链同伦前伸方块中一个链同伦等价诱导出另一个链同伦等价.     

拓扑Markov链——非游荡集和拓扑熵

本文讨论了拓扑Markov链,给出了涉及拓扑熵、紊动点偶、ω-极限集、非游荡集、周期点集和终于周期点集之间关系的7个等价条件。     

树映射具有正拓扑熵的几个等价条件

本文讨论树映射的拓扑熵,得到树映射具有正拓扑熵的几个等价条件。     

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．     

几乎周期性与SS混沌集

引进正则移位不变集的概念,证明了有正则移位不变集的紧致系统在几乎周期点集中存在SS混沌集,特别地,具有正拓扑熵的区间映射在几乎周期点集中存在SS混沌集.     

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.     

From a Floyd-Auslander minimal system to an odd triangular map

In 1989 A.N. Sharkovsky asked the question which of the properties characterizing continuous maps of the interval with zero topological entropy remain equivalent for triangular maps of the square. The problem is difficult and only partial results are known. However, in the case of triangular maps with nondecreasing fibres there are only few gaps in a classification (given by Z. Ko?an) of a set of 24 of these conditions. In the present paper we remove these gaps by giving an example of a triangular map in the square with the following properties:

(1)

all fibre maps are nondecreasing,

(2)

all recurrent points of the map are uniformly recurrent, and

(3)

the restriction of the map to the set of recurrent points has an uncountable scrambled set (and so is Li-Yorke chaotic).

The example is obtained by taking an appropriate Floyd-Auslander minimal system and then taking its appropriate continuous extension to a triangular map of the square.     

华沙圈上连续映射的某些动力性质

本文研究华沙圈上定义的连续映射的动力性质．指出对于定义在华沙圈上的连续自映射而言，有与线段自映射相应的Ｓａｒｋｏｖｓｋｉｉ定理，周期点集的闭包与回归点集的闭包相等，中心为周期点集的闭包，中心的深度不大于４，以及拓扑熵为零的充要条件是它的周期点的周期都是２的方幂．     

The set of recurrent points of a continuous self-map on an interval and strong chaos

In this paper, we discuss a continuous self-map of an interval and the existence of an uncountable strongly chaotic set. It is proved that if a continuous self-map of an interval has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.     

Tree maps having chain movable fixed points

In this paper we discuss some basic properties of chain reachable sets and chain equivalent sets of continuous maps. It is proved that if f:T→T is a tree map which has a chain movable fixed point v, and the chain equivalent set CE(v,f) is not contained in the set P(f) of periodic points of f, then there exists a positive integer p not greater than the number of points in the set End([CE(v,f)])−Pv(f) such that fp is turbulent, and the topological entropy . This result generalizes the corresponding results given in Block and Coven (1986) [2], Guo et al. (2003) [6], Sun and Liu (2003) [10], Ye (2000) [11], Zhang and Zeng (2004) [12]. In addition, in this paper we also consider metric spaces which may not be trees but have open subsets U such that the closures are trees. Maps of such metric spaces which have chain movable fixed points are discussed.     

An Interval Counterexample on Topological Sequence Entropy

We provide an example showing that, in general, the topological sequence entropy of a continuous interval map cannot be attained on the set of non-wandering points. This proves that there is no connection between the topological sequence entropy of an interval map and its behavior on sets of special dynamical meaning.     

The triangular maps with closed sets of periodic points

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F(x,y)=(f(x),gx(y)), satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987-997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.