$\mathbb{R}$- 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 $L^p$, $1\le p<\mathrm{\infty }$. 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.