# On permutable meromorphic functions

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

## Abstract

We study the class of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in , with at least one essential singularity, permutes with a non-constant rational map , then is a Möbius map that is not conjugate to an irrational rotation. For a given function which is not a Möbius map, we show that the set of functions in that permute with ƒ is countably infinite. Finally, we show that there exist transcendental meromorphic functions such that, among functions meromorphic in the plane, ƒ permutes only with itself and with the identity map.

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