An a priori bound for rational functions on the Berkovich projective line
Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 719-738.

On établit une majoration locale a priori pour la dynamique d’une fraction rationnelle f de degré >1 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) μ f de f, sous condition que f 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.

We establish a local a priori bound on the dynamics of a rational function f of degree >1 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 μ f of f, 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.

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DOI : 10.5802/jtnb.1224
Classification : 37P50, 11S82
Mots clés : a priori bound, domaine singulier, equidistribution, moving targets, no potentially good reductions, derivative, Berkovich projective line
Yûsuke Okuyama 1

1 Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Yûsuke Okuyama. An a priori bound for rational functions on the Berkovich projective line. Journal de théorie des nombres de Bordeaux, Tome 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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