拓扑秩为2的邻近Cantor系统是剩余置乱的
Proximal Cantor systems with topological rank 2 are residually scrambled
摘要：
Downarowicz和Maass（2008）提出了所有同胚Cantor极小动力系统的拓扑秩，使用适当有序的Bratteli图。在这项研究中，我们采用这个定义的情况下，所有基本上是最小的零维系统。我们考虑拓扑秩为2且唯一极小集为不动点的情形。Akin和Kolyada（2003）在他们对Li-Yorke灵敏度的研究中表明，如果本质最小系统的唯一最小集是一个不动点，那么该系统必须是邻近的。然而，一个有限的拓扑秩意味着可扩张性;此外，在拓扑秩为2的邻近康托系统的情况下，可扩张性总是从最低阶开始。秩2零维系统可以被认为是在遍历论的广阔领域中考虑的秩1变换的一部分。然而，从拓扑混沌理论的角度来看，这些系统也很有趣;例如，本文证明了所有拓扑秩为2邻近Cantor系统都是剩余扰序的.此外，我们还研究了这些系统上的有限不变测度。显然，这样的系统最多有两个遍历测度。我们给出了这类系统唯一遍历的一个充要条件。此外，我们证明了拓扑混合系统的遍历测度的个数可以是1和2。此外，我们提出的例子，拓扑弱混合，不拓扑混合，和唯一遍历。最后，我们证明了非弱混合系统的遍历测度的个数可以是1和2。
英文摘要：
Downarowicz and Maass (2008) 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 all 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 (2003), in their study of Li--Yorke sensitivity, showed that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. However, a 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 zero-dimensional systems might be thought as a part of the rank 1 transformations that are considered in the vast field of ergodic theory. However, these systems are also interesting from the perspective of topological chaos theory; e.g., in this study, we show that all proximal Cantor systems with topological rank 2 are residually scrambled. In addition, we investigate the finite invariant measures on these systems. Evidently, such systems have at most two ergodic measures. 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 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 systems that are not weakly mixing can be 1 and 2.
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