  In this paper, we show that one can naturally associate a limiting dynamical
system $F: T\longrightarrow T$ on an $\R$-tree to any degenerating sequence of
rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The
construction of $F$ is in $2$ steps: first we use barycentric extension to get
$\E f_n : \Hyp^3 \longrightarrow \Hyp^3$; second, we take appropriate limit on
rescalings of hyperbolic space. An important ingredient we prove here is that
the Lipschitz constant depends only on the degree of the rational map. We show
that the dynamics of $F$ records the limiting length spectra of the sequence
$f_n$.
