A Dynamical Fekete-Szeg\H{o} Theorem

TL;DR

This paper establishes a dynamical analogue of the Fekete-Szeg"H{o} theorem, showing how algebraic polynomials generate Julia sets approximating certain compact sets and connecting heights in arithmetic dynamics.

Contribution

It introduces a dynamical version of the Fekete-Szeg"H{o} theorem, linking algebraic polynomials, Julia sets, and arithmetic heights.

Findings

Julia sets $K_{P_n}$ converge to $Pc(E)$ in Klimek topology

Brolin measures converge to the equilibrium measure $\mu_E$

Rumely height arises as a limit of dynamical heights

Abstract

Let $E\subset\Bbb{C}$ be a compact set symmetric with respect to the real axis. A classical theorem of Fekete-Szeg\H{o} asserts that such a compact set is of logarithmic capacity at least one if and only if it admits approximation by algebraic integers whose Galois conjugates lie arbitrarily close to $E$. In this note we prove a dynamical analogue of this phenomenon. When $\mathrm{cap}(E)=1$, we also show that the algebraic polynomials arising from the Fekete-Szeg\H{o} theorem generate filled Julia sets $K_{P_n}$ which converge to the polynomially convex hull $Pc(E)$ in the Klimek topology, while their Brolin measures converge to the equilibrium measure $\mu_E$. In particular, when $E\subset\Bbb{R}$, this provides a genuine approximation of $E$ by algebraic filled Julia sets. As an arithmetic application, we prove that the Rumely height associated to $E$ arises as a limit of canonical dynamical heights in the sense of Call and Silverman, giving a dynamical counterpart to the equidistribution theorems of Bilu and Rumely.