The Density of Prime Divisors in the Arithmetic Dynamics of Quadratic Polynomials

Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P (f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some non-zero an, and denote the natural density of this set by D(P (f, a0)). The problem of determining D(P (f, a0)) when f is linear has attracted significant study, although it remains unresolved in full generality. In this paper we consider the case of f quadratic, where previously D(P (f, a0)) was known only in a few cases. We show D(P (f, a0)) = 0 regardless of a0 for four infinite families of f , including f = x2 +k, k ∈ Z\{−1}. The proof relies on tools from group theory and probability theory to formulate a sufficient condition for D(P (f, a0)) = 0 in terms of arithmetic properties of the forward orbit of the critical point of f . This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point have been shown to determine many global dynamical properties of a quadratic polynomial. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials.