Sensitivity, proximal extension and higher order almost automorphy

Ye, Xiangdong; Yu, Tao

doi:10.1090/tran/7100

Sensitivity, proximal extension and higher order almost automorphy
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by Xiangdong Ye and Tao Yu PDF

Trans. Amer. Math. Soc. 370 (2018), 3639-3662 Request permission

Abstract:

Let $(X,T)$ be a topological dynamical system, and $\mathcal {F}$ be a family of subsets of $\mathbb {Z}_+$. $(X,T)$ is strongly $\mathcal {F}$-sensitive if there is $\delta >0$ such that for each non-empty open subset $U$ there are $x,y\in U$ with $\{n\in \mathbb {Z}_+: d(T^nx,T^ny)>\delta \}\in \mathcal {F}$. Let $\mathcal {F}_t$ (resp. $\mathcal {F}_{ip}$, $\mathcal {F}_{fip}$) consist of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets).

The following Auslander-Yorke’s type dichotomy theorems are obtained: (1) a minimal system is either strongly $\mathcal {F}_{fip}$-sensitive or an almost one-to-one extension of its $\infty$-step nilfactor; (2) a minimal system is either strongly $\mathcal {F}_{ip}$-sensitive or an almost one-to-one extension of its maximal distal factor; (3) a minimal system is either strongly $\mathcal {F}_{t}$-sensitive or a proximal extension of its maximal distal factor.

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