Non-archimedean normal families

TL;DR

This paper investigates the compactness properties of families of analytic maps in non-Archimedean geometry, establishing convergence results and applications to dynamics on Berkovich projective spaces.

Contribution

It provides new compactness theorems for families of Berkovich analytic maps and applies these to non-Archimedean dynamics, including Fatou set analysis.

Findings

Sequences of analytic maps have subsequences converging pointwise under certain conditions.

Locally, continuous maps become analytic after base change.

Fatou components are hyperbolically embedded in non-Archimedean projective space.

Abstract

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 analytic after a certain base change. We give some applications of these results to the dynamics of an endomorphism f of the projective space. We define the Fatou set as the normality locus of the family of the iterates {f n }. We then generalize to the non-Archimedan setting a theorem of Ueda stating that every Fatou component is hyperbolically imbedded in the projective space.