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A weak Néron model with applications to \(p\)-adic dynamical systems. (English) Zbl 0851.14001

Let \(K\) be a field with a discrete valuation \(v\), \({\mathfrak O}_v\) be the ring of the integers and \(S= \text{Spec } {\mathfrak O}_v\). Let \(V\) be a smooth variety over \(K\) and \(\varphi: V\to V\) be a finite morphism over \(K\). A weak Néron model of \((V, \varphi)\) is a smooth, separated and of finite type \(S\)-scheme \(V\) equipped with a morphism \(\Phi: V\to V\), which extends \(\varphi\) in a sense precised by three axioms.
In this paper the author gives a necessary condition for the existence of a weak Néron model, involving the \(n\)-periodic points of \(\varphi\), and he studies the obstruction to the existence of such a model. It turns out that the obstruction is closely related to a set called the Julia set, defined in the theory of dynamical systems associated with the given morphism \(\varphi\).
In the last section the case \(V= \mathbb{P}^1\) is worked out in detail.

MSC:

14A15 Schemes and morphisms
37-XX Dynamical systems and ergodic theory
14G20 Local ground fields in algebraic geometry
11S85 Other nonanalytic theory
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