Étant donné une fonction rationnelle de degré au moins 2 défini sur un corps de nombres , nous montrons que pour chaque place de , il existe une seule mesure sur l’espace de Berkovich tel que si est un séquence de points de dont les hauteurs -canonique tendent vers zéro, alors les points et leurs -conjugués sont équidistribués selon .
La preuve utilise un relèvement de pour construire une fonction de Arakelov-Green de deux variables pour chaque . La mesure s’obtient comme le laplacien (au sens d’espace de Berkovich) de . Les ingrédients principaux de la preuve sont un principe de minimisation de l’énergie pour et une formule pour le diamètre transfini homogène de l’ensemble rempli de Julia -adique pour chaque place .
Given a rational function on of degree at least 2 with coefficients in a number field , we show that for each place of , there is a unique probability measure on the Berkovich space such that if is a sequence of points in whose -canonical heights tend to zero, then the ’s and their -conjugates are equidistributed with respect to .
The proof uses a polynomial lift of to construct a two-variable Arakelov-Green’s function for each . The measure is obtained by taking the Berkovich space Laplacian of . The main ingredients in the proof are an energy minimization principle for and a formula for the homogeneous transfinite diameter of the -adic filled Julia set for each place .
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
@article{AIF_2006__56_3_625_0, author = {Baker, Matthew H. and Rumely, Robert}, title = {Equidistribution of {Small} {Points,} {Rational} {Dynamics,} and {Potential} {Theory}}, journal = {Annales de l'Institut Fourier}, pages = {625--688}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {3}, year = {2006}, doi = {10.5802/aif.2196}, mrnumber = {2244226}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2196/} }
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 SP - 625 EP - 688 VL - 56 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2196/ DO - 10.5802/aif.2196 LA - en ID - AIF_2006__56_3_625_0 ER -
%0 Journal Article %A Baker, Matthew H. %A Rumely, Robert %T Equidistribution of Small Points, Rational Dynamics, and Potential Theory %J Annales de l'Institut Fourier %D 2006 %P 625-688 %V 56 %N 3 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2196/ %R 10.5802/aif.2196 %G en %F AIF_2006__56_3_625_0
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. http://archive.numdam.org/articles/10.5802/aif.2196/
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