Algebraic Stability for Skew Products

TL;DR

This paper investigates algebraic stability of rational skew products on surfaces, establishing conditions for stability via birational models and employing non-Archimedean dynamics and Berkovich space techniques.

Contribution

It proves algebraic stability for skew products over ruled surfaces when the base map has no superattracting cycles, introducing a new approach using non-Archimedean dynamics.

Findings

Stability achieved via birational morphisms under certain conditions.

Counterexample with superattracting fixed point shows limitations.

Non-Archimedean dynamics and Berkovich space are effective tools.

Abstract

In this article we study algebraic stability for rational skew products in two dimensions $\phi : X \dashrightarrow X$, i.e. maps of the form $\phi(x, y) = (\phi_1(x), \phi_2(x, y))$. We prove that when $X$ is a birationally ruled surface and $\phi_1$ has no superattracting cycles, then we can always find a smooth surface $\hat X$ and an algebraic stabilisation $\pi : (\hat \phi, \hat X) \to (\phi, X)$ which is a birational morphism. We provide an example of a skew product $\phi$ where $\phi_1$ has a superattracting fixed point and $\phi$ 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' $\phi_*: \mathbb P^1_{\text{an}}(\mathbb K) \to \mathbb P^1_{\text{an}}(\mathbb K)$ on the Berkovich projective line over the Puiseux series, $\mathbb K$. The Fatou-Julia theory for $\phi_*$ is instrumental to our approach.