Semiconjugate Rational Functions: A Dynamical Approach
Fedor Pakovich · Arnold Mathematical Journal · 2018
Using dynamical methods we give a new proof of the theorem saying that if A, B, X are rational functions of complex variable z 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.