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 .
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 .
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
Baker, Matthew H. 1; Rumely, Robert 2
@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 = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 56 (2006) no. 3, pp. 625-688. doi : 10.5802/aif.2196. https://aif.centre-mersenne.org/articles/10.5802/aif.2196/
[1] Points entiers sur les surfaces arithmétiques, J. Reine Angew. Math., Volume 531 (2001), pp. 201-235 | DOI | MR | Zbl
[2] Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math., Volume 585 (2005), pp. 61-92 | DOI | MR | Zbl
[3] Harmonic analysis on metrized graphs (to appear in Canad. J. Math.) | Zbl
[4] Spectral theory and analytic geometry over nonarchimedean fields, 33, AMS Mathematical Surveys and Monographs, 1990 (AMS, Providence) | MR | Zbl
[5] Limit distribution of small points on algebraic tori, Duke Math. J., Volume 89 (1997), pp. 465-476 | DOI | MR | Zbl
[6] Subvarieties of linear tori and the unit equation: a survey, Analytic Number Theory (London Math. Soc. Lecture Note), Volume 247 (1996), pp. 1-20 (Kyoto, 1996) | MR | Zbl
[7] Canonical heights on projective space, Journal of Number Theory, Volume 63 (1997), pp. 211-243 | DOI | MR | Zbl
[8] Canonical heights on varieties with morphisms, Compositio Math., Volume 89 (1993), pp. 163-205 | Numdam | MR | Zbl
[9] Equidistribution of small points in finite fibers (preprint available at http://arXiv.org/abs/math.NT/0304023)
[10] Capacity theory on varieties, Compositio Math., Volume 80 (1991), pp. 71-84 | EuDML | Numdam | MR | Zbl
[11] The capacity pairing, J. Reine Angew. Math., Volume 434 (1993), pp. 1-44 | DOI | EuDML | MR | Zbl
[12] Dynamics of rational maps: Lyapunov exponents, bifurcations, and metrics on the sphere, Mathematische Annalen, Volume 326 (2003), pp. 43-73 | DOI | MR | Zbl
[13] Calculus on arithmetic surfaces, Annals of Math., Volume 119 (1984), pp. 387-424 | DOI | MR | Zbl
[14] The valuative tree, Lecture Notes in Mathematics, Volume 1853, Springer-Verlag, Berlin and New York, 2004 | MR | Zbl
[15] Equidistribution des points de petite hauteur (http://arXiv.org/abs/math.NT/0407471, to appear in Math. Annalen)
[16] Théorème d’équidistribution de Brolin en dynamique -adique, C. R. Math. Acad. Sci., Volume 339 (2004) no. 4, pp. 271-276 | MR | Zbl
[17] An invariant measure for rational maps, Bol. Soc. Brasil. Mat., Volume 14 (1983), pp. 45-62 | DOI | MR | Zbl
[18] Superattractive fixed points in , Indiana Univ. Math. J., Volume 43 (1994), pp. 321-365 | DOI | MR | Zbl
[19] Pluripotential Theory, London Mathematical Society Monographs (New Series), 6, Oxford Science Publications, 1991 | MR | Zbl
[20] Arakelov Theory, Springer–Verlag, 1988 | MR | Zbl
[21] Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems, Volume 3 (1983), pp. 351-385 | MR | Zbl
[22] Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, 80, Mémoires de la SMF, Paris, 2000 | Numdam | Zbl
[23] Dynamics in One Complex Variable, Vieweg, 2000 (2nd ed.) | MR | Zbl
[24] Mahler measure for dynamical systems on and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory (Progress in Mathematic), Volume 235 (2004), pp. 219-250 | MR | Zbl
[25] Potential Theory in the Complex Plane, 28, London Math. Soc., 1995 (Student Texts) | MR | Zbl
[26] Théorie de Fatou et Julia dans la droite projective de Berkovich (in preparation)
[27] Dynamique des fonctions rationelles sur les corps locaux, Astérisque, Volume 287 (2003), pp. 147-230 | MR | Zbl
[28] Real Analysis, MacMillan Publishing Co., New York, 1988 (3rd ed.) | MR | Zbl
[29] Real and Complex Analysis, McGraw-Hill, New York, 1974 (2nd edition) | MR | Zbl
[30] Capacity Theory on Algebraic Curves, Lecture Notes in Mathematics, Volume 1378, Springer-Verlag, Berlin-Heidelberg-New York, 1989 | MR | Zbl
[31] An intersection theory for curves, with analytic contributions from nonarchimedean places, Canadian Mathematical Society Conference Proceedings, Volume 15 (1995), pp. 325-357 | MR | Zbl
[32] On Bilu’s equidistribution theorem, Contemp. Math., Volume 237 (1999), pp. 159-166 | MR | Zbl
[33] Analysis and dynamics on the Berkovich projective line (preprint available at http://arXiv.org/abs/math.NT/0407426)
[34] Arithmetic capacities on , Math. Zeit., Volume 215 (1994), pp. 533-560 | DOI | EuDML | MR | Zbl
[35] Existence of the Sectional Capacity, AMS Memoires, 145, American Mathematical Society, Providence, R.I., 2000 no. 690 | MR | Zbl
[36] Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, Berlin and New York, 1994 | MR | Zbl
[37] Équirépartition des petits points, Invent. Math., Volume 127 (1997), pp. 337-347 | DOI | MR | Zbl
[38] Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, University of Rennes (2005) (Ph. D. Thesis)
[39] Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959 (reprinted by Chelsea, New York) | MR | Zbl
[40] Algebra, 1, New York, 1970 | Zbl
[41] Admissible pairing on a curve, Invent. Math., Volume 112 (1993), pp. 171-193 | DOI | MR | Zbl
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