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A finiteness result for common zeros of iterates of rational maps

Chatchai Noytaptim, Xiao Zhong

TL;DR

Let $f$ and $g$ be compositionally independent rational maps with $c$ in $\mathbb{C}(x)$. The paper proves there are only finitely many $\lambda$ with $f^n(\lambda)=g^n(\lambda)=c(\lambda)$ for some $n$, except for explicitly described exceptional conjugacy classes to affine Möbius maps; when at most one map has degree $>1$, the finiteness persists under weaker hypotheses. The strategy combines arithmetic dynamics (canonical heights, Arakelov-Green functions), Diophantine geometry (S-unit theorem, equidistribution of small points), and specialization to transfer results from algebraic settings to $\mathbb{C}$, together with a dynamical Ping-pong analysis to handle the nonlinear case. This work extends GCD-type finiteness phenomena to iterates of rational maps and links rigidity of composition semigroups with arithmetic height methods, providing a structural framework for finiteness results beyond polynomials.

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 $λ\in\mathbb{C}$ with the property that there is an $n$ such that $f^n(λ) = g^n(λ) = c(λ)$, except for a few families of $f, g \in Aut(\mathbb{P}^1_\mathbb{C})$ which gives counterexamples.

A finiteness result for common zeros of iterates of rational maps

TL;DR

Let and be compositionally independent rational maps with in . The paper proves there are only finitely many with for some , except for explicitly described exceptional conjugacy classes to affine Möbius maps; when at most one map has degree , the finiteness persists under weaker hypotheses. The strategy combines arithmetic dynamics (canonical heights, Arakelov-Green functions), Diophantine geometry (S-unit theorem, equidistribution of small points), and specialization to transfer results from algebraic settings to , together with a dynamical Ping-pong analysis to handle the nonlinear case. This work extends GCD-type finiteness phenomena to iterates of rational maps and links rigidity of composition semigroups with arithmetic height methods, providing a structural framework for finiteness results beyond polynomials.

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 are compositionally independent rational functions and , then there are at most finitely many with the property that there is an such that , except for a few families of which gives counterexamples.
Paper Structure (13 sections, 16 theorems, 227 equations)

This paper contains 13 sections, 16 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Historical background.
  3. Statement of main results.
  4. Proof strategy.
  5. Preliminaries
  6. Absolute values over global fields.
  7. Dynamical height and preperiodic points.
  8. Arakelov-Green's function.
  9. Results from Diophantine geometry.
  10. The Ping-pong Lemma
  11. Asymptotic notation
  12. Proof of Theorem \ref{['thm: mainlinear']}
  13. Proof of Theorem \ref{['thm : mainnonlinear']}

Key Result

Theorem 1.3

Let $f(X)$, $g(X)$ be automorphisms on ${\mathbb P}^1$ defined over ${\mathbb C}$ and $c(X)$ be a rational function defined over ${\mathbb C}$. If the semigroup generated by $f(X)$ and $g(X)$ under compositions is free, then there are only finitely many $\lambda \in {\mathbb C}$ such that for some positive integer $n$ unless $f(X)$, $g(X)$ are simultaneously conjugated by an automorphism on ${\ma

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 26 more