The augmented deformation space of rational maps

It was recently shown that the Epstein deformation space of marked rational maps with prescribed combinatorial and dynamical structure can be disconnected. For example, the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical value not lying in this cycle has a deformation space with infinitely many components. We study the structure of the augmented deformation space for this example, and show, in particular, that the closure of deformation space in augmented deformation space is also disconnected in this case.