On the centralizers of endomorphisms of the projective line

TL;DR

This paper proves that for certain dominant endomorphisms of the projective line, the union of their centralizers stabilizes at some finite iterate, confirming a conjecture and extending known results in the dynamics of endomorphisms.

Contribution

It establishes the stabilization of centralizers for non-power map endomorphisms of the projective line, confirming Pakovich's conjecture and extending the Tits alternative to broader semigroup contexts.

Findings

Union of centralizers stabilizes at some finite iterate N.

Confirms a conjecture of F. Pakovich.

Extends Tits alternative to non-finitely generated semigroups.

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 $f$, $C(f^{n}):=\{\textrm{dominant}\;g:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}\;|\; g\circ f^{n}=f^{n}\circ g\}$, $n\geq1$, and prove that their union is equal to $C(f^{N})$ for some $N\geq1$. 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.