A non-archimedean $\lambda$-lemma

TL;DR

This paper develops a non-archimedean version of the λ-lemma for rational functions, establishing a framework for analyzing their dynamics over Berkovich spaces and linking stability conditions.

Contribution

It introduces a non-archimedean λ-lemma and demonstrates the equivalence of stability conditions for rational functions over Berkovich spaces.

Findings

Established a non-archimedean λ-lemma for rational functions.

Proved the equivalence of two stability conditions in the non-archimedean setting.

Provided a framework for studying dynamics over Berkovich spaces.

Abstract

We provide a framework for studying the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over a complete non-archimedean field. We prove a non-archimedean analogue of Ma\~{n}\'{e}, Sad, and Sullivan's $\lambda$-Lemma and use this to show an equivalence of two stability conditions for families of rational functions parametrized by an open subset of the Berkovich affine line.