On permutable meromorphic functions

TL;DR

This paper investigates the permutation properties of meromorphic functions outside countable sets of singularities, revealing conditions under which they commute with rational maps and characterizing their permutation sets.

Contribution

It characterizes when meromorphic functions permute with rational maps and describes the structure of their permutation sets, including existence of functions with minimal permutation partners.

Findings

Functions with essential singularities permute only with certain Möbius maps.

For non-Möbius functions, the set of permuting functions is countably infinite.

Existence of transcendental meromorphic functions that only permute with themselves and the identity.

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 M\"{o}bius map that is not conjugate to an irrational rotation. For a given function $ f \in\mathcal{M}$ which is not a M\"{o}bius 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.