A Tits alternative for endomorphisms of the projective line

Abstract

We prove an analog of the Tits alternative for endomorphisms of $\mathbb{P}^1$. In particular, we show that if $S$ is a finitely generated semigroup of endomorphisms of $\mathbb{P}^1$ over $\mathbb{C}$, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if $f$ and $g$ are polarizable maps over any field of any characteristic and $\operatorname{Prep}(f) \not= \operatorname{Prep}(g)$, then for all sufficiently large $j$, the semigroup $\langle f^j, g^j \rangle$ is a free semigroup on two generators.