Sensitivity, proximal extension and higher order almost automorphy

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. $\mathcal{F}_{ip}$, $\mathcal{F}_{fip}$) be consisting 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.