Commuting rational functions revisited

TL;DR

This paper develops a method to describe all rational functions commuting with a given rational function B, excluding special cases, by constructing a finite group structure and identifying generators through topological methods.

Contribution

It introduces a novel approach to characterize the set of commuting rational functions using group theory and graph-based topological methods, extending previous understanding.

Findings

Defines an equivalence relation on commuting functions

Establishes a finite group structure for the quotient set

Provides generators of the group via fundamental groups of associated graphs

Abstract

Let $B$ be a rational function of degree at least two that is neither a Latt\`es 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 $B$.