A finiteness result for common zeros of iterates of rational maps

Abstract

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 $f,g\in Aut({\mathbb{P}}_{\mathbb{C}}^1)$ which gives counterexamples.

There was an exceptional case missed in Theorem 1.3 due to a minor technical error. It is corrected in the current version and the corresponding updated proof is in page 24-26