Hypercyclic and mixing composition operators on \({\mathscr {O}}_M({\mathbb {R}})\)

In this paper we characterize mixing composition operators acting on the space \({\mathscr {O}}_M({\mathbb {R}})\) ${\mathcal{O}}_M(\mathbb{R})$ of slowly increasing smooth functions. Moreover we relate the mixing property of those operators with the solvability of Abel’s functional equation and we give a sufficient condition for sequential hypercyclicity of composition operators on \({\mathscr {O}}_M({\mathbb {R}})\) ${\mathcal{O}}_M(\mathbb{R})$ . This is used to prove that many mixing composition operators are hypercyclic.