Greatest common divisors of iterates of polynomials

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, a; b is an element of C[x], there is a polynomial h such that for all n, we have gcd(a(n) - 1, b(n) - 1) | h We prove a compositional analog of this theorem, namely that if f; g is an element of C[x] are compositionally independent polynomials and c(x) is an element of C[x], then there are at most finitely many lambda with the property that there is an n such that (x - lambda) divides gcd(f degrees(n)(x) - c(x), g degrees(n)(x) - c(x)) .