Broken family sensitivity in transitive systems

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.