Sensitivity, proximal extension and higher order almost automorphy (1605.01119v2)

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.