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 \(L^p\) , \(1 \le p < \infty \) . 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.
