On quasi-weakly almost periodic points

查看全文 作　　者：HE [1]WeiHong;YIN [2]JianDong;ZHOU [3]ZuoLing

机构地区：[1]School of Mathematics and Computational Science, Sun Yat-sen University;[2]Department of Mathematics, Nanchang University;[3]Lingnan College, Sun Yat-sen University University

出　　处：《Science China Mathematics》索引2013年第56卷第3期,共10页

基　　金：supported by National Natural Science Foundation of China (Grant Nos.10971236 and 11261039);the Foundation from the Jiangxi Education Department (Grant No. GJJ11295);the Natural Science Foundation of Jiangxi Province of China (Grant No. 20114BAB201006);the Foundation of Sun Yat-sen University Advanced Center 摘　　要：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. 关键词：周期点离散动力系统周期轨道渐近行为开放性问题拓扑结构测量中心不变测度

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On quasi-weakly almost periodic points

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