iteration. Hence we introduce the natural action by conjugacy of the set $\Aut(\S):=\Rat_1$ of Moebius transformations on $\Rat_d$ : $$\Aut(\S)\times \Rat_d\ni (M,f)\mapsto M\circ f\circ M^{-1}\in \Rat_d.$$ We want to understand the dynamical properties of $\rat_d$, the quotient of $\Rat_d$ by this action. Here we are going to focus on the study of diverging sequences of $\rat_d$.

Berkovich spaces and the Deligne-Mumford compactification are very general tools that can be used in very general settings but for our purpose we will restrict our use to the completion of the field of formal Puiseux series for Berkovich spaces and to the case of genus zero (moduli space of punctured spheres) for the Deligne-Mumford compactification.

For the Berkovich space, we will recall what is this field later (as late as possible). This will restrict our application of Berkovich spaces in dynamics (for example we will not be able to explain how they are used in [@BD] or in [@FG]).

For the Deligne-Mumford compactification, a new vocabulary with dynamical systems between trees of spheres, that was claimed to be more adapted for the use in holomorphic dynamics, has been introduced in [@A0] (translated in English in [@A1], [@A2], [@A3]). We will directly use it 