Invariant spanning trees for quadratic rational maps

TL;DR

This paper introduces invariant spanning trees and a new combinatorial invariant called the ivy graph to classify quadratic post-critically finite branched coverings, providing a computational method for their analysis.

Contribution

It presents a novel approach using invariant spanning trees and the ivy graph to study Thurston equivalence classes of quadratic maps, along with a computational procedure based on bisets.

Findings

Developed a computational method for finding invariant spanning trees.

Introduced the ivy graph as a new invariant for Thurston classes.

Provided insights into the structure of quadratic post-critically finite branched coverings.

Abstract

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes - the ivy graph.