Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Baker, Matthew H.; Rumely, Robert

doi:10.5802/aif.2196

Equidistribution of Small Points, Rational Dynamics, and Potential Theory
[Équidistribution des petits points, dynamique des fonctions rationnelles, et théorie du potentiel]

Baker, Matthew H.¹ ; Rumely, Robert ²

¹ Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA
² University of Georgia, Department of Mathematics, Athens, GA 30602-7403, USA

Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 625-688.

Résumé
Abstract

Étant donné une fonction rationnelle de degré au moins 2 défini sur un corps de nombres $k$ , nous montrons que pour chaque place $v$ de $k$ , il existe une seule mesure $μ_{ϕ, v}$ sur l’espace de Berkovich $ℙ_{Berk, v}^{1} / ℂ_{v}$ tel que si ${z_{n}}$ est un séquence de points de $ℙ^{1} (\bar{k})$ dont les hauteurs $ϕ$ -canonique tendent vers zéro, alors les points $z_{n}$ et leurs $Gal (\bar{k} / k)$ -conjugués sont équidistribués selon $μ_{ϕ, v}$ .

La preuve utilise un relèvement $F (x, y) = (F_{1} (x, y), F_{2} (x, y))$ de $ϕ$ pour construire une fonction de Arakelov-Green $g_{ϕ, v} (x, y)$ de deux variables pour chaque $v$ . La mesure $μ_{ϕ, v}$ s’obtient comme le laplacien (au sens d’espace de Berkovich) de $g_{ϕ, v} (x, y)$ . Les ingrédients principaux de la preuve sont un principe de minimisation de l’énergie pour $g_{ϕ, v} (x, y)$ et une formule pour le diamètre transfini homogène de l’ensemble rempli de Julia $v$ -adique $K_{F, v} \subset ℂ_{v}^{2}$ pour chaque place $v$ .

Given a rational function $ϕ (T)$ on $ℙ^{1}$ of degree at least 2 with coefficients in a number field $k$ , we show that for each place $v$ of $k$ , there is a unique probability measure $μ_{ϕ, v}$ on the Berkovich space $ℙ_{Berk, v}^{1} / ℂ_{v}$ such that if ${z_{n}}$ is a sequence of points in $ℙ^{1} (\bar{k})$ whose $ϕ$ -canonical heights tend to zero, then the $z_{n}$ ’s and their $Gal (\bar{k} / k)$ -conjugates are equidistributed with respect to $μ_{ϕ, v}$ .

The proof uses a polynomial lift $F (x, y) = (F_{1} (x, y), F_{2} (x, y))$ of $ϕ$ to construct a two-variable Arakelov-Green’s function $g_{ϕ, v} (x, y)$ for each $v$ . The measure $μ_{ϕ, v}$ is obtained by taking the Berkovich space Laplacian of $g_{ϕ, v} (x, y)$ . The main ingredients in the proof are an energy minimization principle for $g_{ϕ, v} (x, y)$ and a formula for the homogeneous transfinite diameter of the $v$ -adic filled Julia set $K_{F, v} \subset ℂ_{v}^{2}$ for each place $v$ .

DOI : 10.5802/aif.2196

Classification : 11G50, 37F10, 31C15
Keywords: Canonical heights, rational dynamics, equidistribution, arithmetic dynamics, potential theory, capacity theory
Mot clés : hauteurs canoniques, dynamique des fonctions rationnelles, équidistribution, dynamique arithmétique, théorie du potentiel

Affiliations des auteurs :

Baker, Matthew H. ¹ ; Rumely, Robert ²

¹ Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA
² University of Georgia, Department of Mathematics, Athens, GA 30602-7403, USA

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     journal = {Annales de l'Institut Fourier},
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Baker, Matthew H.; Rumely, Robert. Equidistribution of Small Points, Rational Dynamics, and Potential Theory. Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 625-688. doi : 10.5802/aif.2196. https://aif.centre-mersenne.org/articles/10.5802/aif.2196/

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