Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Baker, Matthew H.; Rumely, Robert

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

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 ${\mu }_{\varphi ,v}$ sur l’espace de Berkovich ${\mathbb{P}}_{\mathrm{Berk},v}^1/{ℂ}_v$ tel que si $\left\{z_n\right\}$ est un séquence de points de ${\mathbb{P}}^1\left(\overline{k}\right)$ dont les hauteurs $\varphi$-canonique tendent vers zéro, alors les points $z_n$ et leurs $\mathrm{Gal}(\overline{k}/k)$-conjugués sont équidistribués selon ${\mu }_{\varphi ,v}$.

La preuve utilise un relèvement $F(x,y)=(F_1(x,y),F_2(x,y))$ de $\varphi$ pour construire une fonction de Arakelov-Green $g_{\varphi ,v}(x,y)$ de deux variables pour chaque $v$. La mesure ${\mu }_{\varphi ,v}$ s’obtient comme le laplacien (au sens d’espace de Berkovich) de $g_{\varphi ,v}(x,y)$. Les ingrédients principaux de la preuve sont un principe de minimisation de l’énergie pour $g_{\varphi ,v}(x,y)$ 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 $\varphi \left(T\right)$ on ${\mathbb{P}}^1$ 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 ${\mu }_{\varphi ,v}$ on the Berkovich space ${\mathbb{P}}_{\mathrm{Berk},\mathrm{v}}^1/{ℂ}_v$ such that if $\left\{z_n\right\}$ is a sequence of points in ${\mathbb{P}}^1\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\mathrm{Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$.

The proof uses a polynomial lift $F(x,y)=(F_1(x,y),F_2(x,y))$ of $\varphi$ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is obtained by taking the Berkovich space Laplacian of $g_{\varphi ,v}(x,y)$. The main ingredients in the proof are an energy minimization principle for $g_{\varphi ,v}(x,y)$ 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 $v$.

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