Sparsity of stable primes for dynamical sequences

Abstract We show that a dynamical sequence of polynomials over a number field whose set of stable primes is of positive density must necessarily have a very restricted, and, in particular, virtually prosolvable dynamical Galois group. Together with existing heuristics, our results suggest, moreover, that a polynomial all of whose iterates are irreducible modulo a positive density subset of the primes must necessarily be a composition of linear functions, monomials, and Dickson polynomials.