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

Abstract

Let $(X,T)$ be a topological dynamical system and $n\ge 2$. We say that $(X,T)$ is $n$-tuplewise IP-sensitive (resp. $n$-tuplewise thickly sensitive) if there exists a constant $\delta >0$ with the property that for each non-empty open subset $U$ of $X$, there exist $x_1,x_2,\dots ,x_n\in U$ such that \[ \Bigl\{k\in\mathbb{N}\colon \min_{1\le i<j\le n}d(T^k x_i,T^k x_j)>δ\Bigr\} \] is an IP-set (resp. a thick set). We obtain several sufficient and necessary conditions of a dynamical system to be $n$-tuplewise IP-sensitive or $n$-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is $n$-tuplewise IP-sensitive for all $n\ge 2$, while it is $n$-tuplewise thickly sensitive if and only if it has at least $n$ 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${}^{\ast }$-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${}^{\ast }$-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP${}^{\ast }$-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.

19 pages. Minor changes. To appear in Discrete and Continuous Dynamical Systems

References in corpus (1)

• Stronger versions of sensitivity for minimal group actions