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 $\{n\in {\mathbb{Z}}_{+}:d(T^{n}x,T^{n}y)>\delta \}\in \mathcal{F}$. 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.

24 pages, revised version following referees' reports. To appear in Transactions of the AMS

References in corpus (3)

• Mean equicontinuity and mean sensitivity
• Analogues of Auslander-Yorke theorems for multi-sensitivity
• Nil Bohr-sets and almost automorphy of higher order