Disjoint \(\mathscr {F}\)-transitivity and topological multiple recurrence of weighted composition operators on \(H(\mathbb {D})\)

Let \(S(\mathbb {D})\) and \(H(\mathbb {D})\) denote the class of holomorphic self-maps and holomorphic functions on the open unit disk \(\mathbb {D}\) in the complex plane \(\mathbb {C}\), respectively. Given \(\varphi _k\in S(\mathbb {D})\) and \(w_k\in H(\mathbb {D})\) for \(k=1,2,\ldots ,N\), we investigate the disjoint \(\mathscr {F}\)-transitivity of the weighted composition operators \(C_{w_1,\varphi _1},\ldots ,C_{w_N,\varphi _N}\) on \(H(\mathbb {D})\). Moreover, we present a condition on the inducing symbols to ensure the topological multiple recurrence of a single weighted composition operator.