Linear Dynamics of Multiplication and Composition Operators on \(\textrm{Hol}(\mathbb {D})\)

We give a complete description of the linear dynamics of multiplication \(M_m\) $M_m$ and composition operators \(C_\varphi \) $C_{\phi }$ on the space \(\textrm{Hol}(\mathbb {D})\) $\text{Hol}(\mathbb{D})$ of all holomorphic maps on the unit disc. We show that \(M_m\) $M_m$ is never supercyclic, and cyclic if and only if the map m is injective. For composition operators, we prove that if \(\varphi \) $\phi$ has a fixed point in \(\mathbb {D}\) $\mathbb{D}$ , then \(C_\varphi \) $C_{\phi }$ is either not cyclic, or cyclic but not supercyclic on \(\textrm{Hol}(\mathbb {D})\) $\text{Hol}(\mathbb{D})$ . On the other hand, if \(\varphi \) $\phi$ does not have any fixed point in the unit disc, then \(C_\varphi \) $C_{\phi }$ is hypercyclic on \(\textrm{Hol}(\mathbb {D})\) $\text{Hol}(\mathbb{D})$ . We provide explicit expressions of cyclic and hypercyclic vectors. Finally, we make some observations on weighted composition operators on \(\textrm{Hol}(\mathbb {D})\) $\text{Hol}(\mathbb{D})$ .