A finiteness result for common zeros of iterates of rational maps

Answering a question asked by Hsia and Tucker in their paper on the finiteness of greatest common divisors of iterates of polynomials, we prove that if f, g ∈ C(X) are compositionally independent rational functions and c ∈ C(X), then there are at most finitely many λ with the property that there is an n such that fⁿ(λ) = gⁿ(λ) = c(λ), except for a few families of f, g ∈ Aut(P¹C) which gives counterexamples.