A finiteness result for common zeros of iterates of rational maps

TL;DR

This paper proves that for compositionally independent rational functions, the set of points where their iterates share a common value is finite, except for specific families of functions that serve as counterexamples.

Contribution

It establishes a finiteness result for common zeros of iterates of rational maps, answering a question by Hsia and Tucker.

Findings

Finiteness of common zeros for iterates of rational functions.

Counterexamples identified within specific automorphism families.

Extension of previous polynomial gcd finiteness results to rational functions.

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