Algebraic Stability for Skew Products

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.