Family independence for topological and measurable dynamics
Family independence for topological and measurable dynamics
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DOI：
10.1090/s0002-9947-2012-05493-6
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发表时间：
2009-08
影响因子：
1.3
通讯作者：
Wen Huang;Hanfeng Li;X. Ye
中科院分区：
文献类型：
--
作者：
Wen Huang;Hanfeng Li;X. Ye
对于一个族\(F\)（\(Z_+\)的子集的一个集合），\(F\)-独立性的概念在拓扑动力系统（\(t.d.s.\)）和可测动力系统（\(m.d.s.\)）中都有定义。证明了不存在非平凡的\(\{\text{联合的}\}\)-独立的可测动力系统；一个可测动力系统是\(\{\text{正密度}\}\)-独立的当且仅当它具有完全正熵；并且一个可测动力系统是弱混合的当且仅当它是\(\{\text{IP}\}\)-独立的。对于一个拓扑动力系统，证明了不存在非平凡的极小\(\{\text{联合的}\}\)-独立系统；一个拓扑动力系统是弱混合的当且仅当它是\(\{\text{IP}\}\)-独立的。此外，构造了一个非平凡的邻近拓扑\(K\)系统，并且给出了极小拓扑\(K\)蕴含强混合这一事实的一个拓扑证明。
For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.