On permutable meromorphic functions

Aequationes Math. 90 (2016), no. 5, 1025–1034. Also 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.