Quantitative uniform distribution of points of small height on the projective line. (Equidistribution quantitative des points de petite hauteur sur la droite projective.) (English) Zbl 1175.11029

Math. Ann. 335, No. 2, 311-361 (2006); corrigendum 339, No. 4, 799-801 (2007).
Summary: We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of uniform distribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function \(R\) in one variable and defined over a number field \(K\), is associated a normalized height on the algebraic closure of \(K\). We show that these dynamically defined heights are adelic in our sense, and deduce from this uniform distribution results for preimages of points under \(R\) at every place of \(K\). Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over \(\mathbb C_p\), for each prime \(p\).


11G50 Heights
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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[1] Autissier, P.: Points entiers sur les surfaces arithmétiques. J. Reine Angew. Math. 531, 201–235 (2001) · Zbl 1007.11041
[2] Baker, M., Hsia, L.: Canonical Heights, Transfinite Diameters, and Polynomial Dynamics. Preprint disponible à arxiv:math.NT/0305181
[3] Baker, M., Rumely, R.: Equidistribution of small points, rational dynamics and potential theory. Preprint disponible à arxiv:math.NT/0407426 · Zbl 1234.11082
[4] Baker, M., Rumely, R.: Analysis and dynamics on the Berkovich projective line. Preprint disponible à arxiv:math.NT/0407433
[5] Berkovich, V.G.: Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, 33. American Mathematical Society, Providence, RI, 1990 · Zbl 0715.14013
[6] Bilu, Y.: Limit distribution of small points on algebraic tori. Duke Math. J. 89(3), 465–476 (1997) · Zbl 0918.11035
[7] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6, 103–144 (1965) · Zbl 0127.03401
[8] Carleson, L., Gamelin, T.W.: Complex dynamics. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993, x+175 pp · Zbl 0782.30022
[9] Chambert-Loir, A.: Mesures et équidistribution sur les espaces de Berkovich. Prépublication disponible sur le site arxiv:math.NT/0304023 · Zbl 1112.14022
[10] Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., 1507, Springer, Berlin, 1992, pp. 87–104
[11] Denker, M., Przytycki, F., Urbański, M.: On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory Dynam. Systems 16(2), 255–266 (1996) · Zbl 0852.46024
[12] Favre, C., Jonsson, M.: The valuative tree. Lecture Notes in Mathematics, 1853. Springer-Verlag, Berlin, 2004 · Zbl 1064.14024
[13] Favre, C., Rivera-Letelier, J.: Théorème d’équidistribution de Brolin en dynamique p-adique. C. R. Math. Acad. Sci. Paris 339(4), 271–276 (2004) · Zbl 1052.37039
[14] Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14(1), 45–62 (1983) · Zbl 0568.58027
[15] Haydn, N.: Convergence of the transfer operator for rational maps. Ergodic Theory Dynam. Systems 19(3), 657–669 (1999) · Zbl 0953.37006
[16] Hindry, M., Silverman, J.H.: Diophantine geometry, an introduction. Graduate Texts in Mathematics, 201. Springer-Verlag, New York, 2000 · Zbl 0948.11023
[17] Ljubich, M.J.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dynam. Systems 3(3), 351–385 (1983)
[18] Petsche, C.: The distribution of Galois orbits of low height. PhD dissertation. The university of Texas at Austin, 2003
[19] Rivera-Letelier, J.: Théorie de Julia et Fatou sur la droite projective de Berkovich. En préparation
[20] Rumely, R.: On Bilu’s equidistribution theorem. Spectral problems in geometry and arithmetic (Iowa City, IA, 1997), Contemp. Math., 237, Amer. Math. Soc., Providence, RI, 1999, pp. 159–166 · Zbl 1029.11030
[21] Pineiro, J., Szpiro, L., Tucker, T.J.: Mahler measure for dynamical systems on and intersection theory on a singular arithmetic surface. Progress in Math. Birkhauser, 235 (2004). Geometric Methods in Algebra and Number Theory · Zbl 1101.11020
[22] Pollicott, M., Sharp, R.: Large deviations and the distribution of pre-images of rational maps. Comm. Math. Phys. 181(3), 733–739 (1996) · Zbl 0919.30020
[23] Sibony, N.: Dynamique des applications rationnelles de . Dynamique et géométrie complexes (Lyon, 1997), ix–x, xi–xii, 97–185, Panor. Synthèses, 8, Soc. Math. France, Paris, 1999
[24] Szpiro, L., Ullmo, E., Zhang, S.: Equirépartition des petits points. Invent. Math. 127(2), 337–347 (1997) · Zbl 0991.11035
[25] Thuillier, A.: Thèse de l’université de Rennes, 2005
[26] Tsuji, M.: Potential theory in modern function theory. Reprinting of the 1959 original. Chelsea Publishing Co., New York, 1975, x+590 pp. 31–02
[27] Ullmo, E.: Théorie ergodique et géométrie arithmétique. Proceedings of the International Congress of Mathematicians, Vol. II, 197–206, Higher Ed. Press, Beijing 2002
[28] Zhang, S.: Small points and adelic metrics. J. Algebraic Geom. 4(2), 281–300 (1995) · Zbl 0861.14019
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