  We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a
quadratic polynomial with integer coefficients. We show, for four infinite
families of f, that the set of primes dividing at least one term of such a
sequence must have density zero, regardless of choice of a_0. The proof relies
on tools from group theory and probability theory to develop a zero-density
criterion 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
determine many global dynamical properties of f. The article also includes
apparently new work on the irreducibility of iterates of quadratic polynomials.

Number Theory