arXiv subjects

Richard A. P. Birkett

Publications and source records attributed to Richard A. P. Birkett.

4 recordsLinked to original sources

arXiv2026

The Calculus of Blowups on a Ruled Surface

The purposes of this article are threefold. First, to determine numerically when an arbitrary blowup of a smooth surface is smooth. We show the surface is smooth if and only if certain rational parameters involving log discrepancy and multiplicity of the exceptional divisors form a generalised Farey sequence within the dual graph of divisors. Second, in doing the above we provide an exposition of the Berkovich projective line $\mathbb P^1_{\text{an}}(\mathbb K)$ over the Puiseux series as a universal dual graph for divisors on a ruled surface. Third, to explain how non-Archimedean skew products interact with this multiplicity structure of the tree.

math.AG↗

arXiv2024

Algebraic Stability for Skew Products

In this article we study algebraic stability for rational skew products in two dimensions $ϕ: X \dashrightarrow X$, i.e. maps of the form $ϕ(x, y) = (ϕ_1(x), ϕ_2(x, y))$. We prove that when $X$ is a birationally ruled surface and $ϕ_1$ has no superattracting cycles, then we can always find a smooth surface $\hat X$ and an algebraic stabilisation $π: (\hat ϕ, \hat X) \to (ϕ, X)$ which is a birational morphism. We provide an example of a skew product $ϕ$ where $ϕ_1$ 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' $ϕ_*: \mathbb P^1_{\text{an}}(\mathbb K) \to \mathbb P^1_{\text{an}}(\mathbb K)$ on the Berkovich projective line over the Puiseux series, $\mathbb K$. The Fatou-Julia theory for $ϕ_*$ is instrumental to our approach.

math.DS↗

arXiv2024

On the Stabilisation of Rational Surface Maps

The dynamics of a rational surface map $f : X \dashrightarrow X$ are easier to analyse when $f$ is `algebraically stable'. Here we investigate when and how this condition can be achieved by conjugating $f$ with a birational change of coordinates. We show that if this can be done with a birational morphism, then there is a minimal such conjugacy. For birational $f$ we also show that repeatedly lifting $f$ to its graph gives a stable conjugacy. Finally, we give an example in which $f$ can be birationally conjugated to a stable map, but the conjugacy cannot be achieved solely by blowing up. La dynamique d'une application rationnelle $f : X \dashrightarrow X$ sur une surface est plus simple à analyser lorsque $f$ est `algébriquement stable'. Dans cet article nous étudions comment la stabilité peut être réalisée en conjuguant $f$ par un changement de variable birationnel. Nous montrons que si cela peut être réalisée avec une morphisme birationnelle, il existe une telle conjugaison minimale. Pour $f$ birationnelle, nous montrons aussi que l'on obtient une conjugaison stable par relévement successif au graphe. Nous donnons enfin un exemple dans lequel $f$ peut être conjuguée birationnellement à une application stable, mais la conjuguée ne peut pas être obtenue uniquement par éclatement.

math.DS↗

arXiv2023

Skew Products on the Berkovich Projective Line

In this article, we develop a dynamical theory for what shall be called a skew product on the Berkovich projective line, $ϕ_*: \mathbb{P}^1_{\text{an}}(K) \to \mathbb{P}^1_{\text{an}}(K)$ over a non-Archimedean field $K$. These functions are defined algebraically yet strictly generalise the notion of a rational map on $\mathbb{P}^1_{\text{an}}$. We describe the analytical, algebraic, and dynamical properties of skew products, including a study of periodic points, and a Fatou/Julia dichotomy. The article culminates with the classification of the connected components of the Fatou set.

math.DS↗