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 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.
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To Phil Rippon on the occasion of his 65th birthday
D. J. Sixsmith was supported by Engineering and Physical Sciences Research Council grant EP/J022160/1.
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Osborne, J.W., Sixsmith, D.J. On permutable meromorphic functions. Aequat. Math. 90, 1025–1034 (2016). https://doi.org/10.1007/s00010-016-0426-y
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DOI: https://doi.org/10.1007/s00010-016-0426-y