  We present a new compactification $M(d,n)$ of the moduli space of self-maps
of $\mathbb{CP}^1$ of degree $d$ with $n$ markings. It is constructed via GIT
from the stable maps moduli space $\ ar M_{0,n}(\mathbb{CP}^1 \times
\mathbb{CP}^1, (1,d))$. We show that it is the coarse moduli space of a smooth
Deligne-Mumford stack and we compute its rational Picard group. Using the
recursive boundary structure inherited from the stable maps space, we give an
explicit algorithm for computing top-intersection numbers of divisors on
$M(d,n)$. We also study the $m$-fold iteration map $M(d,n) \dashrightarrow
M(d^m,n)$ and we give a geometric way to extend this rational map to parts of
the boundary of $M(d,n)$.
