Quadratic rational maps with integer multipliers

TL;DR

This paper classifies quadratic rational maps with integer multipliers over imaginary quadratic fields, showing they are limited to power, Chebyshev, or Lattès maps, supporting Milnor's conjecture.

Contribution

It proves that such maps are restricted to specific types, providing evidence for a conjecture about rational maps with integer multipliers.

Findings

Quadratic rational maps with integer multipliers are classified as power, Chebyshev, or Lattès maps.

Supports Milnor's conjecture on rational maps with integer multipliers.

Provides a classification result in the context of imaginary quadratic fields.

Abstract

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Latt\`{e}s map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.