Lehto--Virtanen-type and big Picard-type theorems for Berkovich analytic spaces

TL;DR

This paper proves non-archimedean analogues of classical complex theorems, establishing value distribution and hyperbolicity properties for Berkovich spaces and morphisms with essential singularities.

Contribution

It introduces Lehto--Virtanen and big Picard theorems for Berkovich spaces, extending complex value distribution results to non-archimedean geometry.

Findings

Establishment of a Lehto--Virtanen-type theorem for Berkovich disks.

Proof of a big Picard-type theorem for Berkovich projective spaces.

Demonstration of hyperbolicity of Berkovich harmonic Fatou sets.

Abstract

In non-archimedean setting, we establish a Lehto--Virtanen-type theorem for a morphism from the punctured Berkovich closed unit disk $\overline{\mathsf{D}}\setminus\{0\}$ in the Berkovich affine line to the Berkovich projective line $\mathsf{P}^1$ having an isolated essential singularity at the origin, and then establish a big Picard-type theorem for such an open subset $\Omega$ in the Berkovich projective space $\mathsf{P}^N$ of any dimension $N$ that the family of all morphisms from $\overline{\mathsf{D}}\setminus\{0\}$ to $\Omega$ is normal in a non-archimedean Montel's sense. As an application of the latter theorem, we see a big Brody-type hyperbolicity of the Berkovich harmonic Fatou set of an endomorphism of $\mathsf{P}^N$ of degree $>1$.