Null systems and sequence entropy pairs
Null systems and sequence entropy pairs
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DOI：
10.1017/s0143385702001724
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发表时间：
2003-09
影响因子：
0.9
通讯作者：
Wen Huang;S. M. Li;S. Shao;X. Ye
中科院分区：
文献类型：
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作者：
Wen Huang;S. M. Li;S. Shao;X. Ye
如果任何序列的度量（相应的拓扑）序列熵为零，则测度保留变换（相应的拓扑系统）为零。库什尼连科证明，遍历测度保持变换当且仅当它为空时才具有离散谱。我们证明，对于最小系统，该陈述对于模几乎一对一的扩展仍然成立。它使我们能够证明散射系统与任何零极小系统都是不相交的。此外，还得到了传递性非最小系统为零的一些必要条件。本地化序列熵的概念，我们定义序列熵对并表明任何系统都存在最大零因子。同时，我们定义了一个较弱的概念，即弱混合对。事实证明，当且仅当不在对角线中的任何对是序列熵对时，系统才是弱混合，当且仅当弱混合对同样成立时，回答了 Blanchard 等人中的一个问题（F. Blanchard, B. Host 和 A. Maass, 拓扑复杂度. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662）。对于群作用，我们直接证明了由包含区域邻近关系的最小不变等价关系导出的因子是等连续的。此外，我们表明非等连续最小远端系统不是零。
A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained. Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.