Topological complexity, return times and weak disjointness
Topological complexity, return times and weak disjointness
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DOI：
10.1017/s0143385703000543
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发表时间：
2004-05
影响因子：
0.9
通讯作者：
Wen Huang;X. Ye
中科院分区：
文献类型：
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作者：
Wen Huang;X. Ye
在布兰查德等人（《拓扑复杂性，遍历理论与动力系统》20（2000年），641 - 662页）的研究中，作者引入了散射的概念以及一个更弱的2 - 散射概念。这两个概念是否等价是一个未解决的问题。本文对该问题给出了肯定的答案。利用开覆盖沿着一些自然数序列的复杂性函数，我们刻画了温和混合、强散射和散射。我们表明温和混合（分别地，强混合）系统与极小一致刚性（分别地，极小刚性）系统不相交。此外，在极小性的假设下，我们证明一个动力系统是完全散射（分别地，温和混合或弱混合）当且仅当它是强混合（分别地，$IP^*$ - 传递或$\mathcal{D}$ - 传递），其中$\mathcal{D}$是具有下巴拿赫密度为1的$\mathbb{Z}_+$的子集的集合。
In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems. Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or $\mathcal{D}$-transitive), where $\mathcal{D}$ is the collection of subsets of $\mathbb{Z}_+$ with the lower Banach density 1.