Broken family sensitivity in transitive systems

TL;DR

This paper explores broken family sensitivity in topological dynamical systems, characterizing it through essential sensitive tuples and recurrence properties for different Furstenberg families, and analyzing their relation to equicontinuous factors.

Contribution

It provides a comprehensive characterization of broken family sensitivity for various Furstenberg families and links it to recurrence and equicontinuity in dynamical systems.

Findings

Broken $ ext{F}_ ext{ps}$- and $ ext{F}_ ext{pubd}$-sensitivity characterized by essential $n$-sensitive tuples and recurrence.

$ ext{F}_ ext{inf}$-sensitivity characterized by persistent minimal distances among points.

Examples illustrating distinctions between different types of broken family sensitivity.

Abstract

Let $(X,T)$ be a topological dynamical system, $n\geq 2$ and $\mathcal{F}$ be a Furstenberg family of subsets of $\mathbb{Z}_+$. $(X,T)$ is called broken $\mathcal{F}$-$n$-sensitive if there exist $\delta>0$ and $F\in\mathcal{F}$ such that for every opene (non-empty open) subset $U$ of $X$ and every $l\in\mathbb{N}$, there exist $x_1^l,x_2^l,\dotsc,x_n^l\in U$ and $m_l\in \mathbb{Z}_+$ satisfying $d(T^k x_i^l, T^k x_j^l)> \delta,\ \forall 1\leq i<j\leq n, k\in m_l+ F\cap[1,l]$. We investigate broken $\mathcal{F}$-$n$-sensitivity for the family of all piecewise syndetic subsets ($\mathcal{F}_{ps}$), the family of all positive upper Banach density subsets ($\mathcal{F}_{pubd}$) and the family of all infinite subsets ($\mathcal{F}_{inf}$). We show that a transitive system $(X,T)$ is broken $\mathcal{F}$-$n$-sensitive for $\mathcal{F}=\mathcal{F}_{ps}\ \text{or}\ \mathcal{F}_{pubd}$ if and only if there exists an essential $n$-sensitive tuple which is an $\mathcal{F}$-recurrent point of $(X^n, T^{(n)})$; is broken $\mathcal{F}_{inf}$-$n$-sensitive if and only if there exists an essential $n$-sensitive tuple $(x_1,x_2,\dotsc,x_n)$ such that $\limsup_{k\to\infty}\min_{1\leq i<j\leq n}d(T^kx_i,T^kx_j)>0$. We also obtain specific properties for them by analyzing the factor maps to their maximal equicontinuous factors. Furthermore, we show examples to distinguish different kinds of broken family sensitivity.