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