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Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

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Abstract

Let f be a rational function of degree d > 1 on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of f, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for C 1-test functions in terms of the proximity of wandering critical orbits to the initial points, and show that the order is \({O(\sqrt{kd^{-k}})}\) upto a specific exceptional set of capacity 0 of initial points, which is contained in the set of superattracting periodic points and the omega-limit set of wandering critical points from the Julia set or the presingular domains of f. As an application in arithmetic dynamics, together with a dynamical Diophantine approximation, these estimates recover Favre and Rivera-Letelier’s quantitative equidistribution in a purely local manner.

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  1. Yûsuke Okuyama

    Present address: Institut de Mathématiques de Jussieu, UPMC Univ Paris 06, UMR 7586, 4 place Jussieu, 75005, Paris, France

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  1. Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto, 606-8585, Japan

    Yûsuke Okuyama

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  1. Yûsuke Okuyama
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Correspondence to Yûsuke Okuyama.

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Dedicated to Professor Masahiko Taniguchi on his sixtieth birthday

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Okuyama, Y. Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics. Math. Z. 273, 811–837 (2013). https://doi.org/10.1007/s00209-012-1032-x

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