Distributional chaos for weighted translation operators on groups

Abstract: In this paper, we study distributional chaos for weighted translations on locally compact groups. We give a sufficient condition for such operators to be distributionally chaotic and construct an example of distributionally chaotic weighted translations by way of the sufficient condition. In particular, we prove the existence of distributional chaos and Li-Yorke chaos for weighted translations operators with aperiodic elements. Furthermore, we also investigate the set of distributionally irregular vectors ($DIV$) of weighted translations through the cone and equivalence classes. When the field is that of complex numbers, we uncover several properties on certain subsets of $DIV$, including their connectedness and correspondences with some measurable subsets in locally compact groups.