is natural to keep trace of the critical points. Of course, as we are doing dynamics, we will also look at the behavior of periodic orbits. To keep trace of these elements, we “mark” them on the sphere. Given a rational map $f\in\Rat_d$ and a finite set $X$ of elements which are meaningful for us (critical points, orbits,...), we define a sphere marked by $X$ to be an injection $i:X\to\S$.

We are interested in $[f]\in\rat_d$ so the space that we will consider is not the space of injections but its quotient by the corresponding action: the injection $i$ will be considered modulo post composition by a Moebius transformation.

For a finite set $X$ containing at least three elements, the space $\Mod_X$ is the space of injections of $X$ in $\S$ modulo post composition by Moebius transformations and is called the moduli space of marked spheres.

For example, when $\card X=3$ the space $\Mod_X$ has a unique element because the action of the Moebius transformation on the subsets of three points of $\P^1\C$ is transitive. In particular, $\Mod_X$ is compact in this case. For $\card X>3$ this is not the case and this space has been well studied by people in 