On \begin{document}$ n $\end{document}-tuplewise IP-sensitivity and thick sensitivity.

Let (X , T) (X , T) be a topological dynamical system and n ≥ 2 n ≥ 2. We say that (X , T) (X , T) is n n -tuplewise IP-sensitive (resp. n n -tuplewise thickly sensitive) if there exists a constant δ > 0 δ > 0 with the property that for each non-empty open subset U U of X X , there exist x 1 , x 2 , ... , x n ∈ U x 1 , x 2 , ... , x n ∈ U such that { k ∈ N : min 1 ≤ i < j ≤ n d (T k x i , T k x j) > δ } { k ∈ N : min 1 ≤ i < j ≤ n d (T k x i , T k x j) > δ } is an IP-set (resp. a thick set). We obtain several sufficient and necessary conditions of a dynamical system to be n n -tuplewise IP-sensitive or n n -tuplewise thickly sensitive and show that any non-trivial weakly mixing system is n n -tuplewise IP-sensitive for all n ≥ 2 n ≥ 2 , while it is n n -tuplewise thickly sensitive if and only if it has at least n n 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 ∗ ∗ -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 ∗ ∗ -equicontinuous. We show that every minimal system admits a maximal almost pairwise IP ∗ ∗ -equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors. [ABSTRACT FROM AUTHOR]