Generalized rescaling limits of a sequence of rational maps

Abstract

We consider a sequence of complex rational maps (f_n) of a fixed degree d at least 2. Building on the seminal work of Kiwi, we introduce the notion of generalized rescaling limits. These are rational maps possibly defined over a non-Archimedean field obtained by renormalizing at some scale a fixed iterate of the sequence (f_n). We explain that the set of all generalized rescaling limits is naturally organized as a tree, and bound the size of this tree in term of the degree d. We apply our theory to quadratic rational maps. Using Kiwi's classification, we describe all possible trees in this case, and prove a uniform bound on the number of cycles with small multipliers.