On permutable meromorphic functions

Osborne, J. W. and Sixsmith, D. J. (2016). On permutable meromorphic functions. Aequationes Mathematicae, 90(5) pp. 1025–1034.

DOI: https://doi.org/10.1007/s00010-016-0426-y

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 ${f \in \mathcal{M}}$ which is not a Möbius map, we show that the set of functions in ${\mathcal{M}}$ that permute with ƒ 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, ƒ permutes only with itself and with the identity map.

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