Equidistribution of preimages over nonarchimedean fields for maps of good reduction

Gignac, William

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1737-1779.

Dans cet article, nous montrons un analogue du théorème d’équidistribution des préimages en dynamique complexe pour des applications définies sur un corps non archimédien et ayant bonne réduction. Bien que, en général ce théorème ne soit qu’un analogue partiel, nous montrons que pour la plupart des applications ayant bonne réduction c’est un analogue exact. Dans le cas particulier où le corps non archimédien est muni de la norme triviale, nous montrons un résultat plus fort, à savoir que les préimages des valuations «  modérées  » sont équidistribuées asymptotiquement par rapport à une mesure canonique.

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages of any tame valuation equidistribute to a canonical measure.

Classification : 37P50, 37P55, 37P05
Keywords: equidistribution, nonarchimedean dynamics, Berkovich spaces, maps of good reduction, multiplicities, exceptional set.
Mot clés : équidistribution, dynamique non archimédienne, espaces de Berkovich, applications ayant bonne réduction, multiplicités, ensemble exceptionnel

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