| 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 Möbius map that is not conjugate to an irrational rotation. For a given function ({f inmathcal{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. |
