$\mathbb R$- and $\mathbb C$-supercyclicity for some classes of operators

In the present paper we investigate different variants of supercyclicity, precisely $\mathbb R^+$-, $\mathbb R$- and $\mathbb C$-supercyclicity in the context of composition operators. We characterize $\mathbb R$-supercyclic composition operators on $L^p$, $1 \leq p < \infty$. Then, we turn our attention to dissipative composition operators, and we show that $\mathbb R$- and $\mathbb C$-supercyclicity are equivalent notions in this setting and they have a ``shift-like'' characterization.