${\bf F}$ and if for all pair of internal vertices $v$ and $w:=F(v)$, there exists sequences of isomorphisms $M_{n,v}^Y:\S_n^Y\to \S_v$ and $M_{n,w}^Z:\S_n^Z\to \S_w$ such that

-   $M_{n,v}^Y\circ a_n^Y:Y\to \S_v$ converges to $a_v^Y:Y\to \S_v$,

-   $M_{n,w}^Z\circ a_n^Z:Z\to \S_w$ converges to $a_w^Z:Z\to \S_w$ and

-   $M_{n,w}^Z\circ f_n\circ (M_{n,v}^Y )^{-1}:\S_v\to \S_w$ converges locally uniformly outside $Y_v$ to ${f_v:\S_v\to \S_w}$.

These covers have natural properties such as

-   combinatorial surjectivity (every sphere of $\T^Z$ has a preimage),

-   global degree (the global number of preimages of every point on a sphere of $\T^Z$ is constant) and

-   allowing Riemann-Hurwitz type formula for a well chosen equivalent of the Euler characteristic.

Maps on Berkovich space
-----------------------

For every polynomial $P_t:=X^d+a_1.X^{d-1}+\ldots+a_{d}\in \LL[X]$ we have the explicit formula: $$P(B(0,r))=B(a_d(t),\underset{i\in[1,d]}{max}(|a_i(t)|_\LL . r^{d-i})).$$ Hence $P_t$ defines a dynamic on $  \P^1 _{Berk}$.

However the case of rational maps is a little more difficult but we can also associate a unique ball as an image to another one and thus define a dynamic on $\P^1 _{Berk}$. For this we have to remark that the map $F_t$ is well defined on $\P^1\LL$, and that for each branching point$a\in \P^1 _{Berk}$ all the open balls corresponding to the branches of $a$ (see 