Aequationes mathematicae

, Volume 90, Issue 5, pp 1025–1034 | Cite as

On permutable meromorphic functions

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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.

Mathematics Subject Classification

Primary 30D05 Secondary 30D30 
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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe Open UniversityMilton KeynesUK

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