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 $v$.

Classification : 11G50, 37F10, 31C15
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

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

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

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

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

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

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

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

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

[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., Volume 80 (1991), pp. 71-84 | EuDML | Numdam | MR | Zbl

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

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

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

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

[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., Volume 339 (2004) no. 4, pp. 271-276 | MR | Zbl

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

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

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

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

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

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

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

[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 (Progress in Mathematic), Volume 235, Birkhaüser (2004), pp. 219-250 | MR | Zbl

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

[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, Volume 287 (2003), pp. 147-230 | Numdam | MR | Zbl

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

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

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

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

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

[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., Volume 215 (1994), pp. 533-560 | DOI | EuDML | MR | Zbl

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

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

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

[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 Ph.D. Thesis)

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

[40] van der Waerden, B. L. Algebra, 1, New York, 1970 | Zbl

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