Approximability of dynamical systems between trees of spheres

We study sequences of analytic conjugacy classes of rational maps that diverge in moduli space. In particular, we are interested in the notion of rescaling limits introduced by Jan Kiwi. From [A1], we recall the notion of dynamical covers between trees of spheres for which a periodic sphere corresponds to a rescaling limit. We study necessary conditions for such a dynamical cover to be the limit of dynamically marked rational maps. With these conditions, we classify these covers in the case of bi-critical maps, and we recover the second main result of Jan Kiwi regarding rescaling limits.