An a priori bound for rational functions on the Berkovich projective line

Okuyama, Yûsuke

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) ${\mu }_f$ 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 ${\mu }_f$ 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.

Classification : 37P50, 11S82
Mots clés : a priori bound, domaine singulier, equidistribution, moving targets, no potentially good reductions, derivative, Berkovich projective line

[1] Baker, Matthew; Rumely, Robert Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, 159, American Mathematical Society, 2010 | DOI | Zbl

[2] Bedford, Eric; Smillie, John Polynomial diffeomorphisms of ${ℂ}^2$: currents, equilibrium measure and hyperbolicity, Invent. Math., Volume 103 (1991) no. 1, pp. 69-99 | DOI | Zbl

[3] Bedford, Eric; Taylor, B. Alan Fine topology, Šilov boundary, and ${\left(d{d}^c\right)}^n$, J. Funct. Anal., Volume 72 (1987), pp. 225-251 | DOI | Zbl

[4] Benedetto, Robert L. Dynamics in one non-archimedean variable, Graduate Studies in Mathematics, 198, American Mathematical Society, 2019 | DOI

[5] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990

[6] Buff, Xavier; Gauthier, Thomas Quadratic polynomials, multipliers and equidistribution, Proc. Am. Math. Soc., Volume 143 (2015) no. 7, pp. 3011-3017 | DOI | Zbl

[7] Chambert-Loir, Antoine Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl

[8] Fatou, Pierre Sur les équations fonctionnelles, Bull. Soc. Math. Fr., Volume 48 (1920), pp. 33-94 | DOI | Numdam

[9] Favre, Charles; Rivera-Letelier, Juan Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc., Volume 100 (2010) no. 1, pp. 116-154 | DOI | Zbl

[10] Fresnel, Jean; van der Put, Marius Rigid analytic geometry and its applications, Progress in Mathematics, 218, Birkhäuser, 2004 | DOI

[11] Gauthier, Thomas Equidistribution towards the bifurcation current I: multipliers and degree $d$ polynomials, Math. Ann., Volume 366 (2016) no. 1-2, pp. 1-30 | DOI | Zbl

[12] Gauthier, Thomas; Vigny, Gabriel 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] Jonsson, Mattias Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications (Lecture Notes in Mathematics), Volume 2119, Springer, 2015, pp. 205-366 | Zbl

[14] Kawaguchi, Shu; Silverman, Joseph H. Nonarchimedean Green functions and dynamics on projective space, Math. Z., Volume 262 (2009) no. 1, pp. 173-197 | DOI | Zbl

[15] Nevanlinna, Rolf Analytic functions, Grundlehren der Mathematischen Wissenschaften, 162, Springer, 1970 | DOI

[16] Okuyama, Yûsuke 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] Okuyama, Yûsuke 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] Okuyama, Yûsuke 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] Okuyama, Yûsuke; Vigny, Gabriel Value distribution of derivatives in polynomial dynamics, Ergodic Theory Dyn. Syst., Volume 41 (2021) no. 12, pp. 3780-3806 | DOI | Zbl

[20] Thuillier, Amaury 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] Tsuji, Masatsugu Potential theory in modern function theory, Chelsea Publishing, 1975 (reprinting of the 1959 original)