Algebraic Stability for Skew Products

Abstract

In this article we study algebraic stability for rational skew products in two dimensions $ϕ: X \dashrightarrow X$, i.e. maps of the form $ϕ(x, y) = (ϕ_1(x), ϕ_2(x, y))$. We prove that when $X$ is a birationally ruled surface and $ϕ_1$ has no superattracting cycles, then we can always find a smooth surface $\hat X$ and an algebraic stabilisation $π: (\hat ϕ, \hat X) \to (ϕ, X)$ which is a birational morphism. We provide an example of a skew product $ϕ$ where $ϕ_1$ has a superattracting fixed point and $ϕ$ is not algebraically stable on any model. Our techniques involve transforming the stabilisation issue into a combinatorial dynamical problem for a 'non-Archimedean skew product' $ϕ_*: \mathbb P^1_{\text{an}}(\mathbb K) \to \mathbb P^1_{\text{an}}(\mathbb K)$ on the Berkovich projective line over the Puiseux series, $\mathbb K$. The Fatou-Julia theory for $ϕ_*$ is instrumental to our approach.