Commuting rational functions revisited

Abstract

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.