On permutable meromorphic functions

Accepted for publication by the Aequationes mathematicae. Available on the arXiv. This is a joint work with John Osborne.

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 Möbius map that is not conjugate to an irrational rotation.  For a given function $ f \in\mathcal{M}$ which is not a Möbius 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.