Remark \[remopenball\]) but a finite number of them map to open balls corresponding to the branches of a unique branching point of $\P^1 _{Berk}$. This is unfortunately not easy to prove, we would have to prove the following that we admit in this paper (cf [@Juan]).

If an open ball maps to another open ball, then the corresponding branch maps to the other one.

Thus, for a branching point $v$, this define almost everywhere a map $$D_{v}F_t:T_{v} \P^1 _{Berk}\to T_{F_t(v)} \P^1 _{Berk}.$$

cf Remark \[actionMoeb\].

As tangent spaces are related to limits, it is time to look at the meaning in terms of the map $\rho$. Recall that we defined a map $\rho:\LL\to\C$ which is the equivalent of taking the limit $t\to 0$. This map extends to $\rho:\LL[X]\to\C[X]$ by applying $\rho$ to the coefficients. In order to extend it to a map $$\rho:\LL(X)\to\C(X),$$ we have to be more careful and for each $F_t=P_t/Q_t$ with $P_t,Q_t\in \LL[X]$ we first have to simplify $P_t$ and $Q_t$ by the maximal power of $t$ that can be factored for both of them and, then, we can apply $\rho$ to the coefficients. Of course this is the equivalent of taking the limit when $t\to 0$ 