main ideas are often hidden under a heavy formalism that has to be introduced in order to state corrects results.

What if I tell you that in fact these tools behind their respective formalisms are very simple and that they are also related? One could think that explaining the relation would be very painful and that if it is already hard to learn about one of these topics, it would be crazy to try to understand both of them at the same time. But one would be wrong, as this is exactly the goal of this paper:

-   introducing both of these topics at the same time,

-   explaining the main ideas behind them,

-   expliciting the relation between them, and

-   try to be understandable by a beginner in both of these fields.

First let us recall what I mean by holomorphic dynamics. Let us denote by $\S:=\P^1\C$ the Riemann sphere. According to the Uniformization Theorem, every compact surface of genus $0$ with a projective structure is isomorphic to $\S$. For $d\geq 1$, we denote by $\Rat_d$ the set of rational maps $f:{\mathbb S}\to {\mathbb S}$ of degree $d$. We want to understand the behavior of these maps under 