cover.

A rational map marked by ${\bf F}$ is a triple $(f,y,z)$ such that $f\in \Rat_d$, $y:Y\to \S$ and $z:Z\to \S$ are marked spheres, and the following diagram commutes:

$\xymatrix{
 Y \ar[r]^{y}    \ar[d] _{{ F}} &\S  \ar[d]^{f} \\
      Z \ar[r]_{z}  &\S
  }$

with $\deg_{y(a)}f = \deg(a)$ for all $a\in Y$.

Now we want to consider a sequence $(f_n,i_n,j_n)_n$ of rational maps marked by the same portrait ${\bf F}$ and understand how the trees of spheres are going to help us to define an interesting limit to it. The main idea comes from the following lemma:

[@A2]\[postcomp\] Let $(f_n: \S \to  \S )_n$ be a sequence of rational maps of same degree. Then, there exists a subsequence $(f_{n_k})_{n_k}$ and a sequence of Moebius transformations $(M_{n_k})_{n_k}$ such that $ (M_{n_k}\circ f_{n_k})_{n_k}$ converges to a non constant rational map $f$ uniformly outside a finite number of points.

This mean that a sequence of rational map is never really diverging to a constant but its natural image can disappears to another scale. After passing to a subsequence $i_n$ converges to a tree of sphere $\T^Y$. In our case, each sphere $\S_v$ of $\T^Y$ corresponds to a scale $M_{n,v}$ and another map corresponding to this 