Sensitivity, proximal extension and higher order almost automorphy

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 $x,yin U$ with ${ninmathbb{Z}_+: d(T^nx,T^ny)>delta}inmathcal{F}$. 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-Yorkes 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.