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On quasi-weakly almost periodic points 被引量:8
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作者 HE WeiHong YIN JianDong ZHOU ZuoLing 《Science China Mathematics》 SCIE 2013年第3期597-606,共10页
The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) mea... The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit. 展开更多
关键词 weakly almost periodic point quasi-weakly almost periodic point minimal center of attraction
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On Quasi-weakly Almost Periodic Points of Continuous Flows
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作者 Qi YAN Jian Dong YIN 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2019年第2期257-269,共13页
Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of thi... Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper. 展开更多
关键词 Continuous flow WEAKLY ALMOST periodic point quasi-weakly ALMOST periodic point minimal center of ATTRACTION
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A Note on Quasi-weakly Almost Periodic Point
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作者 Qi YAN Jian Dong YIN Tao WANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2015年第4期637-646,共10页
Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597- 606 (2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the e... Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597- 606 (2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493-502 (2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing. 展开更多
关键词 Topological entropy weakly almost periodic point quasi-weakly almost periodic point strongly mixing
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Banach Upper Density Recurrent Points of C^0-flows
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作者 Qi YAN Jian Dong YIN +1 位作者 Ballesteros MARNELLIE Wei Ling WU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2016年第11期1312-1322,共11页
Let X denote a compact metric space with distance d and F : X×R→ X or Ft : X→X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. T... Let X denote a compact metric space with distance d and F : X×R→ X or Ft : X→X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C0-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points. 展开更多
关键词 C0-flow measure centre weakly almost periodic point quasi-weakly almost periodicpoint Banach upper density recurrent point
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TOPOLOGICAL ENTROPY AND IRREGULAR RECURRENCE
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作者 Lenka Obadalov 《Analysis in Theory and Applications》 2012年第4期321-328,共8页
This paper is devoted to problems stated by Z. Zhou and ELi in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map f and its topolo... This paper is devoted to problems stated by Z. Zhou and ELi in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map f and its topological entropy. The negative answer follows by our recent paper. But for continuous maps of the interval and other more general one-dimensional spaces we give more results; in some cases the answer is positive. 展开更多
关键词 topological entropy weakly almost periodic point quasi-weakly almost peri-odic point
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具有弱specification性质的amenable群作用的拟弱几乎周期点的回复层次
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作者 周亚婷 尹建东 《数学进展》 CSCD 北大核心 2023年第5期945-954,共10页
设(X,G)是一个G-系统,{F_(n)}_(n=1)^(∞)是G的一个Fϕlner序列.本文证明了如果(X,G)具有强specification性质,则在(X,G)上存在一个相对于G的Fϕlner序列{F_(n)}_(n=1)^(∞)的真拟弱几乎周期点x和一个由x沿着{F_(n)}_(n=1)^(∞)的轨道生... 设(X,G)是一个G-系统,{F_(n)}_(n=1)^(∞)是G的一个Fϕlner序列.本文证明了如果(X,G)具有强specification性质,则在(X,G)上存在一个相对于G的Fϕlner序列{F_(n)}_(n=1)^(∞)的真拟弱几乎周期点x和一个由x沿着{F_(n)}_(n=1)^(∞)的轨道生成的不变测度μ,使得μ的支撑与x的相对于Fϕlner序列{F_(n)}_(n=1)^(∞)的极小吸引中心相同.此外,如果(X,G)有弱specification性质且(X,G)的周期测度在(X,G)的不变测度集合M(X,G)中稠密,则在(X,G)上存在一个相对于G上的Fϕlner序列{F_(n)}_(n=1)^(∞)的真拟弱几乎周期点y,使得每一个由y的轨道沿Fϕlner序列{F_(n)}_(n=1)^(∞)生成的不变测度v的支撑与点y相对于{F_(n)}_(n=1)^(∞)的极小吸引中心不相等. 展开更多
关键词 specification性质 不变测度 amenable群作用 真拟弱几乎周期点
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真拟弱几乎周期点和拟正则点 被引量:1
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作者 王肖义 何伟弘 黄煜 《中国科学:数学》 CSCD 北大核心 2013年第12期1185-1192,共8页
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... 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. 展开更多
关键词 不变测度 真拟弱几乎周期点 拟正则点 SPECIFICATION 性质
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Level of the orbit's topological structure and topological semi-conjugacy 被引量:5
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作者 周作领 何伟弘 《Science China Mathematics》 SCIE 1995年第8期897-907,共11页
For a discrete system, the idea that the orbit’s topological structure possesses three levels is proposed and the notions of the quasi-weakly almost periodic point and the minimal covering of a topological semi-conju... For a discrete system, the idea that the orbit’s topological structure possesses three levels is proposed and the notions of the quasi-weakly almost periodic point and the minimal covering of a topological semi-conjugacy are introduced. The relationship between the three levels and the recurrence of points and some properties kept under the topological semi-conjugacy is also discussed. 展开更多
关键词 compact system invariant measure LEVEL of the orbit’s TOPOLOGICAL structure quasi-weakly almost periodic point TOPOLOGICAL semi-conjugacy and minimal covering.
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