Family independence for topological and measurable dynamics

Huang, Wen; Li, Hanfeng; Ye, Xiangdong

doi:10.1090/S0002-9947-2012-05493-6

Family independence for topological and measurable dynamics
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by Wen Huang, Hanfeng Li and Xiangdong Ye PDF

Trans. Amer. Math. Soc. 364 (2012), 5209-5242 Request permission

Abstract:

For a family $\mathcal {F}$ (a collection of subsets of $\mathbb {Z}_+$), the notion of $\mathcal {F}$-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.; an m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and an 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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