Let $1\le p<\infty $, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space ${{\ell }^{p}}(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.

In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

The main purpose of this paper is to develop a point-free notion of topological transitivity. First, we define transitive frame maps and transitive completely prime filters in Frm, the category of frames and frame maps. Then we discuss the relationship between these notions in Frm and the notions of topological transitive and transitive points in Top. Finally, we investigate the relationship between transitive frame maps and the existence of transitive completely prime filters.