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

TL;DR

This paper introduces new notions of $n$-tuplewise IP-sensitivity and thick sensitivity in topological dynamical systems, characterizes their properties, and explores their relationships with minimal points, factor maps, and equicontinuity.

Contribution

It defines and analyzes $n$-tuplewise IP-sensitive and thickly sensitive systems, providing necessary and sufficient conditions, and establishes dichotomies and maximal factors related to these sensitivities.

Findings

Any non-trivial weakly mixing system is $n$-tuplewise IP-sensitive for all $n extgreater{}2$.

A system is $n$-tuplewise thickly sensitive if and only if it has at least $n$ minimal points.

Minimal systems are characterized as distal if and only if they are pairwise IP$^*$-equicontinuous.

Abstract

Let $(X,T)$ be a topological dynamical system and $n\geq 2$. 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 $x_1,x_2,\dotsc,x_n\in U$ such that \[ \Bigl\{k\in\mathbb{N}\colon \min_{1\le i<j\le n}d(T^k x_i,T^k x_j)>\delta\Bigr\} \] 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 $n\geq 2$, 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$^*$-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.