Existence and non-existence in common hypercyclicity and its frequent variants

Given a family $\mathcal T = (T_λ)_{λ\inΛ}$ of bounded linear operators acting on the same $F$-space $X$ and indexed by a set of parameters $Λ\subset \mathbb{R}^d$, $d\geq1$, we explore the existence and non-existence of vectors that simultaneously satisfy, for all elements of $\mathcal T$, one of t