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.
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Keywords: a priori bound, domaine singulier, equidistribution, moving targets, no potentially good reductions, derivative, Berkovich projective line
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@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/
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