Non-Archimedean normal families
[Familles normales non-archimédiennes]
Annales de l'Institut Fourier, Online first, 56 p.

Nous présentons plusieurs résultats concernant la compacité de l’espace des morphismes entre espaces analytiques au sens de Berkovich. Nous montrons que sous certaines conditions sur l’espace source, toute suite d’applications analytiques à valeurs dans un espace affinoïde admet une sous-suite qui converge ponctuellement vers une application continue. Nous étudions aussi la classe des applications continues qui apparaissent comme de telles limites. Localement ces applications deviennent analytiques après changement de base. Nos résultats amènent naturellement à la notion de familles normales. Nous donnons quelques applications à la dynamique des endomorphismes de l’espace projectif. Nous introduisons deux notions naturelles d’ensemble de Fatou et généralisons dans le cadre non-Archimédien un théorème de Ueda qui stipule que toute composante de Fatou est hyperboliquement plongée dans l’espace projectif.

We present several results on the compactness of the space of morphisms between analytic spaces in the sense of Berkovich. We show that under certain conditions on the source, every sequence of analytic maps having an affinoid target has a subsequence that converges pointwise to a continuous map. We also study the class of continuous maps that arise in this way. Locally, they turn to be analytic after a certain base change. Our results naturally lead to a definition of normal families. We give some applications to the dynamics of an endomorphism of the projective space. We introduce two natural notions of Fatou set and generalize to the non-Archimedan setting a theorem of Ueda stating that every Fatou component is hyperbolically imbedded in the projective space.

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DOI : https://doi.org/10.5802/aif.3432
Classification : 32P05,  37P50,  32A19
Mots clés : famille normale, espace de Berkovich
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Rodríguez Vázquez, Rita. Non-Archimedean normal families. Annales de l'Institut Fourier, Online first, 56 p.

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