Broken family sensitivity in transitive systems

Let $(X,T)$ be a topological dynamical system, $n\geq 2$ and $\mathcal{F}$ Furstenberg family of subsets $\mathbb{Z}_+$. is called broken $\mathcal{F}$-$n$-sensitive if there exist $\delta>0$ $F\in\mathcal{F}$ such that for every opene (non-empty open) subset $U$ $X$ $l\in\mathbb{N}$, $x_1^l,x_2^l,\dotsc,x_n^l\in U$ $m_l\in \mathbb{Z}_+$ satisfying $d(T^k x_i^l, T^k x_j^l)> \delta,\ \forall 1\leq i0$. also obtain specific properties them by analyzing factor maps to their maximal equicontinuous factors. Furthermore, we examples distinguish different kinds sensitivity.