## 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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### References:

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