Geometric invariant theory gives natural compactifications for the moduli spaces of rational maps. But the iteration maps on these moduli spaces can not extend to the boundary. This is related to the existence of rescaling limits. I will discuss the iteration maps on the moduli spaces of Newton maps and give a complete description of the rescaling limits in this case. Moreover, if we mark the fixed points, we get Deligne-Mumford compactifications of these moduli spaces. I will give a relationship between the D-M compactification and the GIT compactification. Speaker(s): Hongming Nie (Indiana University)