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abstract: 'We explicit the relation between the dynamics the Berkovich projective line over the completion of the field of formal Puiseux series and the space dynamical systems between trees of spheres known to be equivalent to the Deligne-Mumford compactification of the Moduli space of marked spheres.'
author:
- |
    Matthieu Arfeux\
    [matthieu.arfeux@stonybrook.edu]{}
title: 'Berkovich spaces and Deligne-Mumford compactification'
---

[**About this preprint.**]{}This preprint does not expose any new result. It may be completed later in a version that would contain a description of its relation between different other works and also concrete application to dynamics but the author wanted to make it quickly available. The readers are very welcome to mail any comment that could improve this preprint.

Introduction
============

[**Motivations.**]{}

Who in holomorphic dynamics did not hear about Berkovich spaces and about Deligne-Mumford compactification? Indeed these tools appear more and more in the literature these last years, explicitly as in [@K1], [@DF], [@HK], [@S], [@A0] for example or implicitly as in DeMarco-McMullen’s trees [@DM], Shishikura’s trees [@Sh1], etc... It would really be a very long task to list where these tools are used! However, it is often painful for people to be introduced in these topics because the 