On several dynamical properties of shifts acting on directed trees

Abstract This article explores the notions of $\mathcal {F}$ -transitivity and topological $\mathcal {F}$ -recurrence for backward shift operators on weighted $\ell ^p$ -spaces and $c_0$ -spaces on directed trees, where $\mathcal {F}$ represents a Furstenberg family of subsets of $\mathbb {N}_0$ . In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.