Canonical Heights, Transfinite Diameters, and Polynomial Dynamics

TL;DR

This paper establishes a product formula linking transfinite diameters of filled Julia sets for polynomial dynamical systems over number fields and extends Bilu's equidistribution theorem to points with diminishing canonical heights.

Contribution

It introduces a simple product formula connecting Julia set diameters across completions and generalizes equidistribution results for small-height points.

Findings

Proves a product formula relating Julia set diameters over different completions.

Generalizes Bilu's equidistribution theorem for sequences with canonical heights approaching zero.

Provides tools for analyzing polynomial dynamics over number fields.

Abstract

Let phi(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating phi gives rise to a dynamical system and a corresponding canonical height function, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of phi over various completions of K, and we apply this formula to give a generalization of Bilu's equidistribution theorem for sequences of points whose canonical heights tend to zero.