Algebraic Stability for Skew Products

Abstract

In this article we study algebraic stability for rational skew products in two dimensions $\ddot{I}•:X⇢X$, i.e. maps of the form $\ddot{I}•(x,y)=(\ddot{I}{•}_1(x),\ddot{I}{•}_2(x,y))$. We prove that when $X$ is a birationally ruled surface and $\ddot{I}{•}_1$ has no superattracting cycles, then we can always find a smooth surface $\hat{X}$ and an algebraic stabilisation $\ddot{I}€:(\hat{\ddot{I}}•,\hat{X})\to (\ddot{I}•,X)$ which is a birational morphism. We provide an example of a skew product $\ddot{I}•$ where $\ddot{I}{•}_1$ has a superattracting fixed point and $\ddot{I}•$ 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' $\ddot{I}{•}_{\ast }:{\mathbb{P}}_{\text{an}}^1(\mathbb{K})\to {\mathbb{P}}_{\text{an}}^1(\mathbb{K})$ on the Berkovich projective line over the Puiseux series, $\mathbb{K}$. The Fatou-Julia theory for $\ddot{I}{•}_{\ast }$ is instrumental to our approach.

35 pages, 7 figures