$J$-class weighted translations on locally compact groups

A bounded linear operator $T$ on a Banach space $X$ (not necessarily separable) is said to be $J$-class operator whenever the extended limit set, say $J_T(x)$ equals $X$ for some vector $x\in X$. Practically, the extended limit sets localize the dynamical behavior of operators. In this paper, using the extended limit sets we will examine the necessary and sufficient conditions for the weighted translation $T_{a,\omega}$ to be $J$-class on a locally compact group $G$, within the setting of $L^p$-spaces for $1 \leq p < \infty$. Precisely, we delineate the boundary between $J$-class and hypercyclic behavior for weighted translations. Then, we will show that for torsion elements in locally compact groups, unlike the case of non-dense orbits of weighted translations, we have $J_{T_{a,\omega}}(0)=L^p(G)$. Finally, we will provide some examples on which the weighted translation $T_{a,\omega}$ is $J$-class but it fails to be hypercyclic.