Generalized rescaling limits of a sequence of rational maps

TL;DR

This paper introduces generalized rescaling limits for sequences of rational maps, organizing these limits into a tree structure and applying the theory to quadratic maps with bounds on cycles.

Contribution

It extends Kiwi's work by defining generalized rescaling limits over non-Archimedean fields and characterizing their organization as a tree with size bounds.

Findings

The set of all generalized rescaling limits forms a tree structure.

Bound on the size of the rescaling limits tree in terms of degree d.

For quadratic maps, all possible trees are classified and a uniform bound on small multiplier cycles is established.

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.