We establish a local a priori bound on the dynamics of a rational function of degree on the Berkovich projective line over an algebraically closed field of arbitrary characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and deduce an equidistribution result for moving targets towards the equilibrium (or canonical) measure of , under the no potentially good reduction condition. This partly answers a question posed by Favre and Rivera-Letelier. We also obtain an equidistribution on the averaged value distribution of the derivatives of the iterated polynomials.
On établit une majoration locale a priori pour la dynamique d’une fraction rationnelle de degré sur la droite projective de Berkovich sur un corps algébriquement clos de caractéristique quelconque et complet pour une norme non archimédienne non triviale. On en déduit un résultat d’équidistribution pour des cibles mobiles vers la mesure d’équilibre (ou la mesure canonique) de , sous condition que n’a pas de bonnes réductions potentielles. Cela répond en partie à une question posée par Favre et Rivera-Letelier. On obtient aussi un résultat d’équidistribution pour la distribution moyenne de valeurs des dérivées des polynômes itérés.
Revised:
Accepted:
Published online:
Keywords: a priori bound, domaine singulier, equidistribution, moving targets, no potentially good reductions, derivative, Berkovich projective line
Yûsuke Okuyama 1
CC-BY-ND 4.0
@article{JTNB_2022__34_3_719_0,
author = {Y\^usuke Okuyama},
title = {An a priori bound for rational functions on the {Berkovich} projective line},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {719--738},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {34},
number = {3},
year = {2022},
doi = {10.5802/jtnb.1224},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1224/}
}
TY - JOUR AU - Yûsuke Okuyama TI - An a priori bound for rational functions on the Berkovich projective line JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 719 EP - 738 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1224/ DO - 10.5802/jtnb.1224 LA - en ID - JTNB_2022__34_3_719_0 ER -
%0 Journal Article %A Yûsuke Okuyama %T An a priori bound for rational functions on the Berkovich projective line %J Journal de théorie des nombres de Bordeaux %D 2022 %P 719-738 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1224/ %R 10.5802/jtnb.1224 %G en %F JTNB_2022__34_3_719_0
Yûsuke Okuyama. An a priori bound for rational functions on the Berkovich projective line. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 719-738. doi : 10.5802/jtnb.1224. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1224/
[1] Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, 159, American Mathematical Society, 2010 | DOI | Zbl
[2] Polynomial diffeomorphisms of : currents, equilibrium measure and hyperbolicity, Invent. Math., Volume 103 (1991) no. 1, pp. 69-99 | DOI | Zbl
[3] Fine topology, Šilov boundary, and , J. Funct. Anal., Volume 72 (1987), pp. 225-251 | DOI | Zbl
[4] Dynamics in one non-archimedean variable, Graduate Studies in Mathematics, 198, American Mathematical Society, 2019 | DOI
[5] Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990
[6] Quadratic polynomials, multipliers and equidistribution, Proc. Am. Math. Soc., Volume 143 (2015) no. 7, pp. 3011-3017 | DOI | Zbl
[7] Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl
[8] Sur les équations fonctionnelles, Bull. Soc. Math. Fr., Volume 48 (1920), pp. 33-94 | DOI | Numdam
[9] Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc., Volume 100 (2010) no. 1, pp. 116-154 | DOI | Zbl
[10] Rigid analytic geometry and its applications, Progress in Mathematics, 218, Birkhäuser, 2004 | DOI
[11] Equidistribution towards the bifurcation current I: multipliers and degree polynomials, Math. Ann., Volume 366 (2016) no. 1-2, pp. 1-30 | DOI | Zbl
[12] Distribution of points with prescribed derivative in polynomial dynamics, Riv. Mat. Univ. Parma (N.S.), Volume 8 (2017) no. 2, pp. 247-270 | Zbl
[13] Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications (Lecture Notes in Mathematics), Volume 2119, Springer, 2015, pp. 205-366 | Zbl
[14] Nonarchimedean Green functions and dynamics on projective space, Math. Z., Volume 262 (2009) no. 1, pp. 173-197 | DOI | Zbl
[15] Analytic functions, Grundlehren der Mathematischen Wissenschaften, 162, Springer, 1970 | DOI
[16] Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics, Acta Arith., Volume 161 (2013) no. 2, pp. 101-125 | DOI | Zbl
[17] Effective divisors on the projective line having small diagonals and small heights and their application to adelic dynamics, Pac. J. Math., Volume 280 (2016) no. 1, pp. 141-175 | DOI | Zbl
[18] Value distribution of the sequences of the derivatives of iterated polynomials, Ann. Acad. Sci. Fenn., Math., Volume 42 (2017) no. 2, pp. 563-574 | DOI | Zbl
[19] Value distribution of derivatives in polynomial dynamics, Ergodic Theory Dyn. Syst., Volume 41 (2021) no. 12, pp. 3780-3806 | DOI | Zbl
[20] Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, Ph. D. Thesis, Université Rennes 1 (France) (2005)
[21] Potential theory in modern function theory, Chelsea Publishing, 1975 (reprinting of the 1959 original)
Cited by Sources: