when it is meaningful.

Now we are ready to state the most important statement about functions acting on the Berkovich space.

\[thmzeta\] If $F_t(\zeta)=\zeta$, the tangent map $D_{\zeta}F_t:T_{\zeta} \P^1 _{Berk}\to T_{\zeta} \P^1 _{Berk}$ is a well defined non constant rational map and satisfies $$D_{\zeta}F_t=\rho(F_t).$$

Hence we have the following corollary.

If $deg(\rho(P_t)/\rho(Q_t))>0$ and $(F_t)_{t\in D(0,1)-\{0\}} $ is a holomorphic family in $\Rat_d$ with $F_t:\P^1_\LL\to\P^1_\LL$, then $$F_t=\frac{P_t}{Q_t}\;{\longrightarrow}\;\frac{\rho(P_t)}{\rho(Q_t)}=\rho(F_t).$$

In fact the convergence is local, uniform outside a finite set.

Comparison {#comparison}
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Consider an holomorphic family $$f_t(z):=\frac{a_d(t)z^d+\ldots+a_1(t)z+a_0(t)}{b_d(t)z^d+\ldots+b_1(t)z+b_0(t)},\quad t\in D({0,1}),$$ such that $f_t\in\Rat_d$ if and only if $t\neq0$.

Fix a portrait ${\bf F}$ of degree $d$ and suppose that for $t\neq 0$, we can associate to $F_t$ two marqued spheres $i_t$ and $j_t$ as in the section \[compspace\] and such that $(\F_t,i_t,j_t)$ is a rational map marked by ${\bf F}$. Then $i_t$ and $j_t$ converge respectively to some trees of spheres $\T^Y$ and $\T^Z$ that can be identified in $\P^1_{Berk}$ and, according to Theorem \[toscover\], there is a trees of spheres cover $\F:\T^Y\to\T^Z$.

Given a vertex $v\in T^Y$, there exist a moebius transformation $M_{v,t}$ that maps $\zeta$ to $v$ and $M_{F(v),t}$ the maps $\zeta$ to $F(v)$. Hence the map $$F _{v,t}:=M_{F(v),t}\circ F_t\circ M_{v,t}$$ 