in the same branch if and only if $M_t\circ i_t(a_t)$ and $M_t\circ i_t(b_t)$ have same limit.

\[actionMoeb\] We can define an action of the space of Moebius transformations of $\P^1\LL$ on the branching points of $\P^1_{Berk}$. Indeed, given a branching point $v$, it separates at least three ends of the tree that we can identify to points of $\LL$. Then there exists a unique Moebius transformation $M_t$ that send these three points to $0,1$ and $\infty$. We define the image of $\zeta$ by $M_t$ to be $v$ and this is sufficient to define a transitive action on the all space of branching points of $\P^1_{Berk}$.

Because of this and as we saw in Remark \[zoomstof\] that a sequence (or a family in this setting) of Moebius transformations can be interpret as a rescaling, we can think of the set of branching points of $\P^1_{Berk}$ as a space of all the possible zooms or rescalings. In fact we could go much further and explicit a relation between these Moebius transformations and blows-up.

Maps {#Chap3}
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Trees of spheres covers
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We want to introduce maps between trees of spheres. For this it is convenient to first look at this notion in 