In studying a dynamical system, one finds that there frequently exist some kinds of disturbance or false phenomenon. In order to remove them, we introduce a concept of the measure centre and in order to determine the ...In studying a dynamical system, one finds that there frequently exist some kinds of disturbance or false phenomenon. In order to remove them, we introduce a concept of the measure centre and in order to determine the structure of the measure centre, we also introduce another concept of the weakly almost periodic point. We totally determine the structure of the measure centre and exhibit an example to show that the measure centre may be contained properly in the motion centre and there is a system which is chaotic on the nonwandering set but has zero topological entropy.展开更多
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
In this note, we give the definition of the minimal centre of attraction for self-mapping on a compact metrizable space (for the case of flow, see Ref. [1]) and prove that the minimal centre of attraction coincides wi...In this note, we give the definition of the minimal centre of attraction for self-mapping on a compact metrizable space (for the case of flow, see Ref. [1]) and prove that the minimal centre of attraction coincides with the measure centre. Let (X, d) be a compact metrizable space and f: X→X be continuous.展开更多
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
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
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
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
基金the National Education Foundation of Chinathe National Basic Research Project "Nonlinear Science".
文摘In studying a dynamical system, one finds that there frequently exist some kinds of disturbance or false phenomenon. In order to remove them, we introduce a concept of the measure centre and in order to determine the structure of the measure centre, we also introduce another concept of the weakly almost periodic point. We totally determine the structure of the measure centre and exhibit an example to show that the measure centre may be contained properly in the motion centre and there is a system which is chaotic on the nonwandering set but has zero topological entropy.
基金the Foundation of Guangdong Province and the Foundation of Advanced Research Zhongshan University
文摘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.
基金This work was supported by the National Basic Research Project "Nonlinear Science" and the National Natural Science Foundation of China
文摘In this note, we give the definition of the minimal centre of attraction for self-mapping on a compact metrizable space (for the case of flow, see Ref. [1]) and prove that the minimal centre of attraction coincides with the measure centre. Let (X, d) be a compact metrizable space and f: X→X be continuous.
基金Supported by National Natural Science Foundation of China(Grant No.11261039)National Natural Science Foundation of Jiangxi Province(Grant No.20132BAB201009)the Innovation Fund Designated for Graduate Students of Jiangxi Province
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant No.11661054)
文摘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.
基金Supported in part by the grant SGS/15/2010 from the Silesian University in Opava
文摘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.
基金Supported by National Natural Science Foundations of China(Grant Nos.11261039,11661054)National Natural Science Foundation of Jiangxi(Grant No.20132BAB201009)
文摘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.
