# Equidistribution of Small Points, Rational Dynamics, and Potential Theory

• [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
• Volume: 56, Issue: 3, page 625-688
• ISSN: 0373-0956

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## Abstract

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Given a rational function $\varphi \left(T\right)$ 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 ${\mu }_{\varphi ,v}$ on the Berkovich space ${ℙ}_{\mathrm{Berk},\mathrm{v}}^{1}/{ℂ}_{v}$ such that if $\left\{{z}_{n}\right\}$ is a sequence of points in ${ℙ}^{1}\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the ${z}_{n}$’s and their $\mathrm{Gal}\left(\overline{k}/k\right)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$.The proof uses a polynomial lift $F\left(x,y\right)=\left({F}_{1}\left(x,y\right),{F}_{2}\left(x,y\right)\right)$ of $\varphi$ to construct a two-variable Arakelov-Green’s function ${g}_{\varphi ,v}\left(x,y\right)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is obtained by taking the Berkovich space Laplacian of ${g}_{\varphi ,v}\left(x,y\right)$. The main ingredients in the proof are an energy minimization principle for ${g}_{\varphi ,v}\left(x,y\right)$ 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 $v$.

## How to cite

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Baker, Matthew H., and Rumely, Robert. "Equidistribution of Small Points, Rational Dynamics, and Potential Theory." Annales de l’institut Fourier 56.3 (2006): 625-688. <http://eudml.org/doc/10160>.

@article{Baker2006,
abstract = {Given a rational function $\varphi (T)$ 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 $\mathbb\{P\}^1_\{\rm Berk,v\} / \mathbb\{C\}_v$ such that if $\lbrace z_n \rbrace$ is a sequence of points in $\mathbb\{P\}^1(\overline\{k\})$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\{\rm Gal\}(\overline\{k\}/k)$-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 $K_\{F,v\} \subset \mathbb\{C\}_v^2$ for each place $v$.},
affiliation = {Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA; University of Georgia, Department of Mathematics, Athens, GA 30602-7403, USA},
author = {Baker, Matthew H., Rumely, Robert},
journal = {Annales de l’institut Fourier},
keywords = {Canonical heights; rational dynamics; equidistribution; arithmetic dynamics; potential theory; capacity theory; canonical heights},
language = {eng},
number = {3},
pages = {625-688},
publisher = {Association des Annales de l’institut Fourier},
title = {Equidistribution of Small Points, Rational Dynamics, and Potential Theory},
url = {http://eudml.org/doc/10160},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Baker, Matthew H.
AU - Rumely, Robert
TI - Equidistribution of Small Points, Rational Dynamics, and Potential Theory
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 625
EP - 688
AB - Given a rational function $\varphi (T)$ 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 $\mathbb{P}^1_{\rm Berk,v} / \mathbb{C}_v$ such that if $\lbrace z_n \rbrace$ is a sequence of points in $\mathbb{P}^1(\overline{k})$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their ${\rm Gal}(\overline{k}/k)$-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 $K_{F,v} \subset \mathbb{C}_v^2$ for each place $v$.
LA - eng
KW - Canonical heights; rational dynamics; equidistribution; arithmetic dynamics; potential theory; capacity theory; canonical heights
UR - http://eudml.org/doc/10160
ER -

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