  Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of
degree $d\geq 2$, and $\mu_f$ the measure of maximal entropy for $f\in Rat_d$.
The map of measures $f\mapsto\mu_f$ is known to be continuous on $Rat_d$, and
it is shown here to extend continuously to the boundary of $Rat_d$ in
$\bar{Rat}_d \simeq {\bf P}^{2d+1}$, except along a locus $I(d)$ of codimension
$d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map
$f\mapsto f^n$ for every $n\geq 2$. The limiting measures are given explicitly,
away from $I(d)$. The degenerations of rational maps are also described in
terms of metrics of non-negative curvature on the Riemann sphere: the limits
are polyhedral.
