2024/08/05 by Birkett, Richard A. P.
#14J26 #32H50 (Secondary) #37F80 #37P50 (Primary) #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2408.02658
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) = (ϕ1(x), ϕ2(x, y)). We prove that when X is a birationally ruled surface and ϕ1 has no superattracting cycles, then we can always find a smooth surface X and an algebraic stabilisation π: ( ϕ, X) → (ϕ, X) which is a birational morphism. We provide an example of a skew product ϕ where ϕ1 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 P1an(\mathbb K) → \mathbb P1an(\mathbb K) on the Berkovich projective line over the Puiseux series, \mathbb K. The Fatou-Julia theory for ϕ_* is instrumental to our approach.