Dynamics on Trees of Spheres

TL;DR

This paper introduces the concept of dynamically marked rational maps and trees of spheres to analyze diverging sequences of rational maps, recovering results on rescaling limits and providing a new framework for understanding their dynamics.

Contribution

It develops the notion of dynamical covers between trees of spheres, linking periodic spheres to rescaling limits and extending Kiwi's results on these limits.

Findings

Introduced dynamical covers between trees of spheres

Linked periodic spheres to rescaling limits

Extended Kiwi's results on rescaling limits

Abstract

We introduce the notion of dynamically marked rational maps. 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 order to deal with those, we introduce the notion of dynamical covers between trees of spheres for which a periodic sphere corresponds to a rescaling limit. We then recover results of Jan Kiwi regarding rescaling limits.