Rational maps with rational multipliers

TL;DR

This paper proves that rational maps with multipliers in a fixed number field are limited to power, Chebyshev, or Lattès maps, confirming a conjecture by Milnor and extending recent results.

Contribution

It establishes a classification of rational maps with multipliers in a number field, showing they are only power, Chebyshev, or Lattès maps, thus strengthening Milnor's conjecture.

Findings

Rational maps with multipliers in a number field are classified as power, Chebyshev, or Lattès maps.

Confirms Milnor's conjecture for rational maps with integer multipliers.

Extends recent proofs by Ji and Xie to a broader setting.

Abstract

In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Latt\`{e}s map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.