Riccati equations and polynomial dynamics over function fields

Given a function field KK and ϕ∈K[x]\phi \in K[x], we study two finiteness questions related to iteration of ϕ\phi: whether all but finitely many terms of an orbit of ϕ\phi must possess a primitive prime divisor, and whether the Galois groups of iterates of ϕ\phi must have finite index in their natural overgroup Aut⁡(Td)\operatorname {Aut}(T_d), where TdT_d is the infinite tree of iterated preimages of 00 under ϕ\phi. We focus particularly on the case where KK has characteristic pp, where far less is known. We resolve the first question in the affirmative for a large (in particular, Zariski-dense) subset of the space of degree-dd polynomials. The main step in the proof is to rule out certain algebraic relations among points in backwards orbits; these relations are given by a type of first-order differential equation called a Riccati equation. We then apply our result on primitive prime divisors and adapt a method of Looper to produce new families of polynomials in every characteristic for which the second question has an affirmative answer. We also prove that almost all quadratic polynomials over Q(t)\mathbb {Q}(t) have iterates whose Galois group is all of Aut⁡(Td)\operatorname {Aut}(T_d).