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 $f, g \in \mathbb{ C}(X)$ are compositionally independent rational functions and $c \in \mathbb{ C}(X)$, then there are at most finitely many $\lambda\in\mathbb{ C}$ with the property that there is an $n$ such that $f^n(\lambda) = g^n(\lambda) = c(\lambda)$, except for a few families of $f, g \in Aut(\mathbb{ P}^1_\mathbb{ C})$ which gives counterexamples.
Categories: math.DS math.NT