On the centralizers of endomorphisms of the projective line

Abstract

Let $f$ be a dominant endomorphism of the projective line, which is not conjugate to a power map $z\mapsto z^{\pm d}$. We consider the centralizers of the iterates of $f$, $C(f^n):=\{\text{dominant}g:{\mathbb{P}}^1\to {\mathbb{P}}^1|g\circ f^n=f^n\circ g\}$, $n\ge 1$, and prove that their union is equal to $C(f^N)$ for some $N\ge 1$. This solves a conjecture of F. Pakovich. As an application, we obtain a Tits alternative for cancellative semigroups of endomorphisms of the projective line, without an assumption of finite generation, extending the results of J.P. Bell, K. Huang, W. Peng and T.J. Tucker.