Dynamics of weighted composition operators on spaces of continuous functions

Our study is focused on the dynamics of weighted composition operators defined on a locally convex space $E\hookrightarrow (C(X),\tau_p)$ with $X$ being a topological Hausdorff space containing at least two different points and such that the evaluations $\{\delta_x:\ x\in X\}$ are linearly independent in $E'$. We prove, when $X$ is compact and $E$ is a Banach space containing a nowhere vanishing function, that a weighted composition operator $C_{\varphi,\omega}$ is never weakly supercyclic on $E$. We also prove that if the symbol $\varphi$ lies in the unit ball of $A(\mathbb{D})$, then every weighted composition operator can never be $\tau_p$-supercyclic neither on $C(\mathbb{D})$ nor on the disc algebra $A(\mathbb{D})$. Finally, we obtain Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.