Rank 2 proximal Cantor systems are residually scrambled
Rank 2 proximal Cantor systems are residually scrambled
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DOI：
10.1080/14689367.2017.1360251
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发表时间：
2018-04
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影响因子：
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通讯作者：
T. Shimomura
中科院分区：
文献类型：
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作者：
T. Shimomura
摘要 Downarowicz 和 Maass [7] 使用正确排序的 Bratteli 图提出了所有同胚 Cantor 最小动力系统的拓扑等级。在本研究中，我们将这个定义应用于本质上最小的零维系统的情况。我们考虑拓扑等级为 2 且唯一最小集为不动点的情况。 Akin 和 Kolyada [2] 已经证明，如果本质上最小系统的唯一最小集是一个不动点，那么该系统必定是邻近的。有限的拓扑等级意味着可扩展性；此外，在拓扑等级为2的近端康托系统的情况下，扩张性总是从最低的程度开始。 Rank 2 近端康托系统是残余扰乱的。我们为这些系统的独特遍历性提供了一个充分必要条件。此外，我们还证明了拓扑混合系统的遍历测度的数量可以是 1 和 2。此外，我们还给出了拓扑弱混合、非拓扑混合和唯一遍历的示例。最后，我们证明非弱混合系统的遍历测度的数量可以是 1 和 2。
ABSTRACT Downarowicz and Maass [7] proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of the essentially minimal zero-dimensional systems. We consider the cases in which topological ranks are 2 and unique minimal sets are fixed points. Akin and Kolyada [2], had shown that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. The finite topological rank implies expansiveness; furthermore, in the case of proximal Cantor systems with topological rank 2, the expansiveness is always from the lowest degree. Rank 2 proximal Cantor systems are residually scrambled. We present a necessary and sufficient condition for the unique ergodicity of these systems. In addition, we show that the number of ergodic measures of the systems that are topologically mixing can be 1 and 2. Moreover, we present examples that are topologically weakly mixing, not topologically mixing, and uniquely ergodic. Finally, we show that the number of ergodic measures of the systems that are not weakly mixing can be 1 and 2.