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arXiv · 1603.07497

On permutable meromorphic functions

J. W. Osborne·D. J. Sixsmith
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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.

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J. W. Osborne, D. J. Sixsmith. 2016-07-14. On permutable meromorphic functions. https://doi.org/10.1007/s00010-016-0426-y

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