the case of marked spheres. Recall that the motivation to look at marked spheres was to mark critical points and periodic points. Hence, if we wanna consider sequences of rational maps and consequently sequences of marked spheres, it is clear that all of the maps in the same sequence should have the same number of critical points and the same number of other marked points, ie the same combinatorial datas.

We fix two finite sets $Y$ and $Z$ with at least three elements. We define a portrait ${\bf F}$ of degree $d\geq 2$ to be a couple $(F,\deg)$ where

-   $F:Y\to Z$ is a map between two finite sets $Y$ and $Z$ and

-   $\deg:Y\to \N-\{0\}$ is a function that satisfies $$\sum_{a\in Y}\bigl(\deg(a) -1\bigr) = 2d-2\quad\text{and}\quad \sum_{a\in F^{-1}(b)} \deg(a) = d\quad\text{ for all } b\in \Z.$$

Typically, $Z\subset \S$ is a finite set, $F:Y\to Z$ is the restriction of a rational map $F:\S\to \S$ to $Y:=F^{-1}(Z)$ and $\deg(a)$ is the local degree of $F$ at $a$. In this case, the Riemann-Hurwitz formula and the conditions on the function $\deg$ implies that $Z$ contains the set $V_F$ of the critical values of $F$ in order to let $F:\S-Y\to \S-Z$ be a 