Dynamics on trees of spheres

We introduce the notion of dynamically marked rational map and study sequences of analytic conjugacy classes of rational maps which diverge in the moduli space. In particular, we are interested in the notion of rescaling limits introduced by Jan Kiwi. For this purpose, we introduce the notions of trees of spheres, covers between them and dynamical covers between them. We will study fundamental properties of these objects. We prove that they appear naturally as limits of marked spheres, respectively marked rational maps and dynamically marked rational maps. We also prove that a periodic sphere in a dynamical cover between trees of spheres corresponds to a rescaling limit. We recover as a byproduct a result of Jan Kiwi regarding the bound on the number of rescaling limits that are not post-critically finite.