Approximability of dynamical systems between trees of spheres

TL;DR

This paper investigates the limits of sequences of rational maps in moduli space, focusing on rescaling limits and dynamical covers between trees of spheres, providing classifications and necessary conditions for such limits.

Contribution

It introduces necessary conditions for dynamical covers to be limits of rational maps and classifies these for bicritical maps, extending Kiwi's results on rescaling limits.

Findings

Classified dynamical covers for bicritical maps.

Established necessary conditions for limits of rational maps.

Reproduced Kiwi's main results on rescaling limits.

Abstract

We study sequences of analytic conjugacy classes of rational maps which diverge in moduli space. In particular, we are interested in the notion of rescaling limits introduced by Jan Kiwi. In the continuity of [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 them for the case of bicritical maps and we recover the second main result of Jan Kiwi regarding rescaling limits.