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arXiv · 0908.0574

Family-independence for topological and measurable dynamics

Wen Huang·Hanfeng Li·Xiangdong Ye
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Abstract

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.

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Wen Huang, Hanfeng Li, Xiangdong Ye. 2009-08-05. Family-independence for topological and measurable dynamics. https://arxiv.org/abs/0908.0574

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