On semiconjugate rational functions

TL;DR

This paper studies the structure of semiconjugate rational functions, revealing that such pairs are either related through iteration or can be characterized using orbifolds with non-negative Euler characteristic.

Contribution

It provides a classification of semiconjugate rational functions, connecting their structure to iterative processes or orbifold descriptions.

Findings

Semiconjugate rational functions are either iteratively related or described by orbifolds.

The paper characterizes solutions using orbifolds of non-negative Euler characteristic.

It advances understanding of functional equations involving rational functions.

Abstract

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of such functions, then either $B$ can be obtained from $A$ by a certain iterative process, or $A$ and $B$ can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.