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,bεℂℂ[x], there is a polynomial h such that for all n, we have gcd(aⁿ-1,bⁿ-1)|h We prove a compositional analog of this theorem, namely that if f,gεℂℂ[x] are compositionally independent polynomials and c(x)εℂℂ[x], then there are at most finitely many λ with the property that there is an n such that (x-λ) divides gcd(f^on(x)-c(x),g^on(x)-c(x)).