Two New Recurrent Levels for C0-Flows
Two New Recurrent Levels for C0-Flows
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DOI：
10.1007/s10440-012-9681-7
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发表时间：
2012-04
影响因子：
1.6
通讯作者：
Yu Huang;Zuoling Zhou
中科院分区：
文献类型：
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作者：
Yu Huang;Zuoling Zhou
动力系统的中心问题是轨道的渐近行为或拓扑结构。然而，只有具有一定重复性并形成一组完整测度的点的轨道才是真正重要的。当然，这样的集合希望尽可能小（在集合包含的意义上）。在本文中，我们讨论这样两个集合：弱几乎周期点的集合和拟弱几乎周期点的集合。虽然这两个集合的定义彼此不同，但我们证明它们的闭包都与动力系统的测量中心（或最小吸引力中心）一致。一般来说，一个点可以具有三个层次的轨道结构：由该点生成的不变测度的支持、其最小吸引力中心及其ω极限集。我们研究了弱几乎周期点和准弱几乎周期点的三级轨道结构。我们证明了准弱几乎周期点具有特别丰富的拓扑轨道结构。我们还提出了一个点属于其自身最小吸引中心的充分必要条件。
The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and itsω-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.