A Tits alternative for endomorphisms of the projective line
Jason P. Bell, Keping Huang, Wayne Peng, Thomas J. Tucker · Journal of the European Mathematical Society · 2023
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) ot= \operatorname{Prep}(g) , then for all sufficiently large j , the semigroup \langle f^j, g^j \rangle is a free semigroup on two generators.