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.
Heights and arithmetic dynamics over finitely generated fields
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abstract
We develop a theory of vector-valued heights and intersections defined relative to finitely generated extensions K/k. These generalize both number field and geometric heights. When k is Q or F_p, or when a non-isotriviality condition holds, we obtain Northcott-type results. We then prove a version of the Hodge Index Theorem for vector-valued intersections, and use it to prove a rigidity theorem for polarized dynamical systems over any field.
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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.