On $ n $-tuplewise IP-sensitivity and thick sensitivity

Abstract

Let \begin{document}$ (X,T) $\end{document} be a topological dynamical system and \begin{document}$ n\geq 2 $\end{document}. We say that \begin{document}$ (X,T) $\end{document} is \begin{document}$ n $\end{document}-tuplewise IP-sensitive (resp. \begin{document}$ n $\end{document}-tuplewise thickly sensitive) if there exists a constant \begin{document}$ \delta>0 $\end{document} with the property that for each non-empty open subset \begin{document}$ U $\end{document} of \begin{document}$ X $\end{document}, there exist \begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document} such that

\begin{document}$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i\delta\Bigr\} $\end{document}

is an IP-set (resp. a thick set).

We obtain several sufficient and necessary conditions of a dynamical system to be \begin{document}$ n $\end{document}-tuplewise IP-sensitive or \begin{document}$ n $\end{document}-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is \begin{document}$ n $\end{document}-tuplewise IP-sensitive for all \begin{document}$ n\geq 2 $\end{document}, while it is \begin{document}$ n $\end{document}-tuplewise thickly sensitive if and only if it has at least \begin{document}$ n $\end{document} minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP\begin{document}$ ^* $\end{document}-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP\begin{document}$ ^* $\end{document}-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP\begin{document}$ ^* $\end{document}-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.