Finitely generated semigroups of endomorphisms of P1 over characteristic 0 that have exponential growth have uniform exponential growth, with diameter of independence at most 2 when a degree at least 2 map exists.
Fakhruddin, Questions on self maps of algebraic varieties, J
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A uniform Tits alternative for endomorphisms of the projective line
Finitely generated semigroups of endomorphisms of P1 over characteristic 0 that have exponential growth have uniform exponential growth, with diameter of independence at most 2 when a degree at least 2 map exists.