Equidistribution of Small Points, Rational Dynamics, and Potential Theory
Annales de l'Institut Fourier, Volume 56 (2006) no. 3, pp. 625-688.

Given a rational function ϕ(T) on 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 μ ϕ,v on the Berkovich space Berk ,v 1 / v such that if {z n } is a sequence of points in 1 (k ¯) whose ϕ-canonical heights tend to zero, then the z n ’s and their Gal (k ¯/k)-conjugates are equidistributed with respect to μ ϕ,v .

The proof uses a polynomial lift F(x,y)=(F 1 (x,y),F 2 (x,y)) of ϕ to construct a two-variable Arakelov-Green’s function g ϕ,v (x,y) for each v. The measure μ ϕ,v is obtained by taking the Berkovich space Laplacian of g ϕ,v (x,y). The main ingredients in the proof are an energy minimization principle for g ϕ,v (x,y) and a formula for the homogeneous transfinite diameter of the v-adic filled Julia set K F,v v 2 for each place v.

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 μ ϕ,v sur l’espace de Berkovich Berk ,v 1 / v tel que si {z n } est un séquence de points de 1 (k ¯) dont les hauteurs ϕ-canonique tendent vers zéro, alors les points z n et leurs Gal (k ¯/k)-conjugués sont équidistribués selon μ ϕ,v .

La preuve utilise un relèvement F(x,y)=(F 1 (x,y),F 2 (x,y)) de ϕ pour construire une fonction de Arakelov-Green g ϕ,v (x,y) de deux variables pour chaque v. La mesure μ ϕ,v s’obtient comme le laplacien (au sens d’espace de Berkovich) de g ϕ,v (x,y). Les ingrédients principaux de la preuve sont un principe de minimisation de l’énergie pour g ϕ,v (x,y) et une formule pour le diamètre transfini homogène de l’ensemble rempli de Julia v-adique K F,v v 2 pour chaque place v.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2196
Classification: 11G50,  37F10,  31C15
Keywords: Canonical heights, rational dynamics, equidistribution, arithmetic dynamics, potential theory, capacity theory
@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},
     zbl = {pre05176555},
     mrnumber = {2244226},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2196/}
}
TY  - JOUR
TI  - Equidistribution of Small Points, Rational Dynamics, and Potential Theory
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 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/
UR  - https://zbmath.org/?q=an%3Apre05176555
UR  - https://www.ams.org/mathscinet-getitem?mr=2244226
UR  - https://doi.org/10.5802/aif.2196
DO  - 10.5802/aif.2196
LA  - en
ID  - AIF_2006__56_3_625_0
ER  - 
%0 Journal Article
%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://doi.org/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] 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 Tome 33, AMS Mathematical Surveys and Monographs, 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 (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 | EuDML: 90115 | 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 | EuDML: 153478 | 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, Tome 1853, Springer-Verlag, Berlin and New York, 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 n , Indiana Univ. Math. J., Tome 43 (1994), pp. 321-365 | Article | MR: 1275463 | Zbl: 0858.32023

[19] Klimek, M. Pluripotential Theory, London Mathematical Society Monographs (New Series), Tome 6, Oxford Science Publications, 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 Tome 80, Mémoires de la SMF, Paris, 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. Mahler measure for dynamical systems on 1 and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory (Progress in Mathematic) Tome 235 (2004), pp. 219-250 | MR: 2166086 | Zbl: 1101.11020

[25] Ransford, T. Potential Theory in the Complex Plane Tome 28, London Math. Soc., 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: 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, Tome 1378, Springer-Verlag, Berlin-Heidelberg-New York, 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, 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 n , Math. Zeit., Tome 215 (1994), pp. 533-560 | Article | EuDML: 174627 | MR: 1269489 | Zbl: 0794.31008

[35] Rumely, R.; Lau, C.F.; Varley, R. Existence of the Sectional Capacity, AMS Memoires, Tome 145, American Mathematical Society, Providence, R.I., 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 (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 Tome 1, New York, 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

Cited by Sources: