Algebraic Stability and Skew Products on the Berkovich Projective Line

The dynamics of a rational surface map φ: X ⇢ X are easier to analyse when φ is 'algebraically stable'. In this thesis, we discuss algebraic stability and develop tools from non−Archimedean dynamics in order to understand the algebraic stability of skew product maps. There are three main outcomes, separated into four chapters. First, we investigate when and how algebraically stability can be achieved by conjugating φ with a birational change of coordinate. Such a conjugacy is called an '*algebraic stabilisation*'. We show that if this can be done with a birational morphism, then there is a minimal such stabilisation. For birational φ; we also show that repeatedly lifting φ to its graph gives an algebraic stabilisation. We provide an example in which φ can be birationally conjugated to a stable map, but the conjugacy cannot be achieved solely by blowing up. Second, we develop a dynamical theory for what we call a 'skew product' on the Berkovich projective line, φ\_\*: ℙ¹(K) ⇢ ℙ¹(K). Skew products strictly generalise the notion of a '*rational map*' on ℙ¹(K). We describe the analytical, algebraic, and dynamical properties of skew products, including a study of periodic points, and a Fatou/Julia dichotomy. The chapter ends with the classification of the connected components of the Fatou set. Third we consider algebraic stabilisation for a skew product on a ruled surface φ: X ⇢ X classically this is a mapping of the form φ(x,y) = (φ₁(x), φ₂(x,y)). The main result is that when φ₁ has no superattracting cycles, then we can always find an algebraic stabilisation. We provide an example of a skew product φ where φ₁ has a superattracting fixed point and φ is not algebraically stable in any model. Any product φ: X ⇢ X of this kind induces one φ\_\*: ℙ¹(𝕂) ⇢ ℙ¹(𝕂) on the Berkovich projective line over the field 𝕂 of Puiseux series; this φ\_\* describes φ on the completion of a fibre of X_b. The Fatou/Julia theory for φ\_\* is instrumental to understand algebraic stabilisation for φ. One chapter is devoted to specialising these non−Archimedean tools to the case where K = 𝕂 and describing the transfer of geometric information from one world and skew product to another. We include a description of free and satellite blowups in the dual graph of divisors on X_b.