An algebraic characterization of expanding Thurston maps

TL;DR

This paper provides algebraic criteria to determine when a postcritically finite branched covering map on the sphere is homotopic to an expanding map, enhancing understanding of their topological and dynamical properties.

Contribution

It introduces necessary and sufficient algebraic conditions for homotopy to an expanding map, advancing the classification of Thurston maps.

Findings

Algebraic conditions characterize homotopy to expanding maps

Criteria apply to postcritically finite branched coverings

Results aid in classifying Thurston maps

Abstract

Let $f: S^2 \to S^2$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its postcritical set, to an expanding map $g$.