Semiconjugate rational functions: a dynamical approach

TL;DR

This paper employs dynamical methods to prove that certain rational functions with specific compositional properties have Galois closures of genus zero or one, advancing understanding in complex dynamics and algebraic function theory.

Contribution

It provides a new dynamical proof of a theorem relating rational functions' compositional structure to the genus of their Galois closures.

Findings

Galois closure of the field extension has genus zero or one under given conditions

Dynamical methods offer a new proof of the theorem

Connects rational function composition with algebraic geometry properties

Abstract

Using dynamical methods we give a new proof of the theorem saying that if $A,B,X$ are rational functions of degree at least two such that $A\circ X=X\circ B$ and $\mathbb C(B,X)=\mathbb C(z)$, then the Galois closure of the field extension $\mathbb C(z)/\mathbb C(X)$ has genus zero or one.