Equidistribution of Small Points, Rational Dynamics, and Potential Theory

[Équidistribution des petits points, dynamique des fonctions rationnelles, et théorie du potentiel]

Baker, Matthew H. ; Rumely, Robert

Annales de l'Institut Fourier, Tome 56 (2006), p. 625-688 / Harvested from Numdam

Résumé

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 $\mathbb{P}^1_{{\rm Berk},v} / \mathbb{C}_v$ tel que si $\lbrace z_n \rbrace $ est un séquence de points de $\mathbb{P}^1(\overline{k})$ dont les hauteurs $\varphi $-canonique tendent vers zéro, alors les points $z_n$ et leurs ${\rm Gal}(\overline{k}/k)$-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 $K_{F,v} \subset \mathbb{C}_v^2$ pour chaque place $v$.

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 $v$.

Détail

Publié le : 2006-01-01

MR 2244226 | Zbl pre05176555 | 3 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2196
Classification:  11G50,  37F10,  31C15
Mots clés: hauteurs canoniques, dynamique des fonctions rationnelles, équidistribution, dynamique arithmétique, théorie du potentiel

Bibliographie

[1] Autissier, P. Points entiers sur les surfaces arithmétiques, J. Reine Angew. Math., Tome 531 (2001), pp. 201-235 | Article | MR 1810122 | Zbl 1007.11041

[2] Baker, M.; Hsia, L. C. Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math., Tome 585 (2005), pp. 61-92 | Article | MR 2164622 | Zbl 1071.11040

[3] Baker, M.; Rumely, R. Harmonic analysis on metrized graphs (to appear in Canad. J. Math.) | Zbl 1123.43006

[4] Berkovich, V. G. Spectral theory and analytic geometry over nonarchimedean fields, AMS Mathematical Surveys and Monographs Tome 33 (1990) (AMS, Providence) | MR 1070709 | Zbl 0715.14013

[5] Bilu, Y. Limit distribution of small points on algebraic tori, Duke Math. J., Tome 89 (1997), pp. 465-476 | Article | MR 1470340 | Zbl 0918.11035

[6] Bombieri, E. Subvarieties of linear tori and the unit equation: a survey, Analytic Number Theory, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note) Tome 247 (1996), pp. 1-20 ((Kyoto, 1996)) | MR 1694981 | Zbl 0923.11097

[7] Call, G.; Goldstine, S. Canonical heights on projective space, Journal of Number Theory, Tome 63 (1997), pp. 211-243 | Article | MR 1443758 | Zbl 0895.14006

[8] Call, G.; Silverman, J. Canonical heights on varieties with morphisms, Compositio Math., Tome 89 (1993), pp. 163-205 | Numdam | MR 1255693 | Zbl 0826.14015

[9] Chambert-Loir, A. Equidistribution of small points in finite fibers (preprint available at http://arXiv.org/abs/math.NT/0304023)

[10] Chinburg, T. Capacity theory on varieties, Compositio Math., Tome 80 (1991), pp. 71-84 | Numdam | MR 1127060 | Zbl 0761.11028

[11] Chinburg, T.; Rumely, R. The capacity pairing, J. Reine Angew. Math., Tome 434 (1993), pp. 1-44 | Article | MR 1195689 | Zbl 0756.14013

[12] Demarco, L. Dynamics of rational maps: Lyapunov exponents, bifurcations, and metrics on the sphere, Mathematische Annalen, Tome 326 (2003), pp. 43-73 | Article | MR 1981611 | Zbl 1032.37029

[13] Faltings, G. Calculus on arithmetic surfaces, Annals of Math., Tome 119 (1984), pp. 387-424 | Article | MR 740897 | Zbl 0559.14005

[14] Favre, C.; Jonsson, M. The valuative tree, Lecture Notes in Mathematics, Springer-Verlag, Berlin and New York, Tome 1853 (2004) | MR 2097722 | Zbl 1064.14024

[15] Favre, C.; Rivera-Letelier, J. Equidistribution des points de petite hauteur (http://arXiv.org/abs/math.NT/0407471, to appear in Math. Annalen)

[16] Favre, C.; Rivera-Letelier, J. Théorème d’équidistribution de Brolin en dynamique $p$-adique, C. R. Math. Acad. Sci., Tome 339 (2004) no. 4, pp. 271-276 | MR 2092012 | Zbl 1052.37039

[17] Freire, A.; Lopes, A.; Mañé, R. An invariant measure for rational maps, Bol. Soc. Brasil. Mat., Tome 14 (1983), pp. 45-62 | Article | MR 736568 | Zbl 0568.58027

[18] Hubbard, J. H.; Papadapol, P. Superattractive fixed points in $\mathbb{C}^n$, Indiana Univ. Math. J., Tome 43 (1994), pp. 321-365 | Article | MR 1275463 | Zbl 0858.32023

[19] Klimek, M. Pluripotential Theory, Oxford Science Publications, London Mathematical Society Monographs (New Series), Tome 6 (1991) | MR 1150978 | Zbl 0742.31001

[20] Lang, S. Arakelov Theory, Springer–Verlag (1988) | MR 969124 | Zbl 0667.14001

[21] Lyubich, M. Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems, Tome 3 (1983), pp. 351-385 | MR 741393 | Zbl 0537.58035

[22] Maillot, V. Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, Mémoires de la SMF, Paris Tome 80 (2000) | Numdam | Zbl 0963.14009

[23] Milnor, J. Dynamics in One Complex Variable, Vieweg (2000) (2nd ed.) | MR 1721240 | Zbl 0972.30014

[24] Pineiro, J.; Szpiro, L.; Tucker, T.; Bogomolov, F.; Tschinkel, Y. Mahler measure for dynamical systems on $\mathbb{P}^1$ and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory, Birkhaüser (Progress in Mathematic) Tome 235 (2004), pp. 219-250 | MR 2166086 | Zbl 1101.11020

[25] Ransford, T. Potential Theory in the Complex Plane, London Math. Soc. Tome 28 (1995) (Student Texts) | MR 1334766 | Zbl 0828.31001

[26] Rivera-Letelier, J. Théorie de Fatou et Julia dans la droite projective de Berkovich (in preparation)

[27] Rivera-Letelier, J. Dynamique des fonctions rationelles sur les corps locaux, Astérisque, Tome 287 (2003), pp. 147-230 | MR 2040006 | Zbl 1140.37336 | Zbl 02066305

[28] Royden, H. L. Real Analysis, MacMillan Publishing Co., New York (1988) (3rd ed.) | MR 1013117 | Zbl 0704.26006

[29] Rudin, W. Real and Complex Analysis, McGraw-Hill, New York (1974) (2nd edition) | MR 344043 | Zbl 0278.26001

[30] Rumely, R. Capacity Theory on Algebraic Curves, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, Tome 1378 (1989) | MR 1009368 | Zbl 0679.14012

[31] Rumely, R. An intersection theory for curves, with analytic contributions from nonarchimedean places, Canadian Mathematical Society Conference Proceedings, AMS, Tome 15 (1995), pp. 325-357 | MR 1353942 | Zbl 0859.11037

[32] Rumely, R. On Bilu’s equidistribution theorem, Contemp. Math., Tome 237 (1999), pp. 159-166 | MR 1710794 | Zbl 1029.11030

[33] Rumely, R.; Baker, M. Analysis and dynamics on the Berkovich projective line (preprint available at http://arXiv.org/abs/math.NT/0407426)

[34] Rumely, R.; Lau, C.F. Arithmetic capacities on $\mathbb{P}^n$, Math. Zeit., Tome 215 (1994), pp. 533-560 | Article | MR 1269489 | Zbl 0794.31008

[35] Rumely, R.; Lau, C.F.; Varley, R. Existence of the Sectional Capacity, American Mathematical Society, Providence, R.I., AMS Memoires, Tome 145 (2000) no. 690 | MR 1677934 | Zbl 0987.14018

[36] Silverman, J. Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, Berlin and New York (1994) | MR 1312368 | Zbl 0911.14015

[37] Szpiro, L.; Ullmo, E.; Zhang, S. Équirépartition des petits points, Invent. Math., Tome 127 (1997), pp. 337-347 | Article | MR 1427622 | Zbl 0991.11035

[38] Thuillier, A. 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] Tsuji, M. Potential Theory in Modern Function Theory, Maruzen, Tokyo (1959) (reprinted by Chelsea, New York) | MR 114894 | Zbl 0087.28401

[40] Van Der Waerden, B. L. Algebra, New York Tome 1 (1970) | Zbl 0137.25403

[41] Zhang, S. Admissible pairing on a curve, Invent. Math., Tome 112 (1993), pp. 171-193 | Article | MR 1207481 | Zbl 0795.14015