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We present a new compactification $M(d,n)$ of the moduli space of self-mapsof $\mathbb{CP}^1$ of degree $d$ with $n$ markings. It is constructed via GITfrom 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 smoothDeligne-Mumford stack and we compute its rational Picard group. Using therecursive boundary structure inherited from the stable maps space, we give anexplicit algorithm for computing top-intersection numbers of divisors on$M(d,n)$. We also study the $m$-fold iteration map $M(d,n) \dashrightarrowM(d^m,n)$ and we give a geometric way to extend this rational map to parts ofthe boundary of $M(d,n)$.

A compactification of the moduli space of self-maps of $\mathbb{CP}^1$ using stable maps