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 $\mathrm{Prep}(f)\ne \mathrm{Prep}(g)$, then for all sufficiently large $j$, the semigroup $⟨f^j,g^j⟩$ is a free semigroup on two generators.