Compactification and trees of spheres covers

TL;DR

This paper discusses the topology of the space of covers between trees of spheres, showing how to make this space compact by quotienting out isomorphisms, which is important for understanding convergence in dynamical systems.

Contribution

It introduces a topology on the space of covers between trees of spheres that ensures compactness after quotienting by isomorphisms, extending previous convergence concepts.

Findings

The topology aligns with the convergence notion from prior work.

The space of covers becomes sequentially compact under this topology.

Quotienting by isomorphisms yields a compact space.

Abstract

We already saw in [A1] that the space of dynamically marked rational maps can be identified to a subspace of the space of covers between trees of spheres on which there is a notion of convergence that makes it sequentially compact. In the following we describe a topology on this space quotiented by the natural action of its group of isomorphisms. This topology corresponds to the previous convergence notion and makes this space compact.