# The Arakelov-Zhang pairing and Julia sets

## TL;DR

This paper explores the Arakelov-Zhang pairing between rational maps, providing explicit formulas involving Julia sets, and applies these results to height bounds and polynomial reduction properties.

## Contribution

It derives a simplified expression for the pairing with power maps in terms of Julia sets and establishes bounds on height differences and polynomial reduction.

## Key findings

- Explicit formula for the pairing with power maps involving Julia sets
- Bounds on the difference between canonical and Weil heights
- Rigidity results for polynomials with strong good reduction

## Abstract

The Arakelov-Zhang pairing $\langle\psi,\phi\rangle$ is a measure of the "dynamical distance" between two rational maps $\psi$ and $\phi$ defined over a number field $K$. It is defined in terms of local integrals on Berkovich space at each completion of $K$. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height $h_\phi$ and the standard Weil height $h$, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.