On permutable meromorphic functions

Abstract

We study the class $\mathcal{M}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in $\mathcal{M}$, with at least one essential singularity, permutes with a non-constant rational map $g$, then $g$ is a M\"{o}bius map that is not conjugate to an irrational rotation. For a given function $ f \in\mathcal{M}$ which is not a M\"{o}bius map, we show that the set of functions in $\mathcal{M}$ that permute with $ f $ is countably infinite. Finally, we show that there exist transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C}$ such that, among functions meromorphic in the plane, $f$ permutes only with itself and with the identity map.