Algorithmic aspects of branched coverings IV/V. Expanding maps

TL;DR

This paper characterizes Thurston maps that are isotopic to expanding maps using combinatorial and algebraic methods, providing a decomposition algorithm and applying results to polynomial matings.

Contribution

It introduces a new decomposition framework for Thurston maps into Levy-free and finite-order parts, with an algorithmic approach and applications to polynomial matings.

Findings

Thurston maps decompose uniquely into Levy-free and finite-order components.

The decomposition is algorithmically computable.

Expanding polynomial matings are characterized by the absence of periodic ray cycles.

Abstract

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as maps with "contracting biset". We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure. We apply these results to matings of post-critically finite polynomials, extending a criterion by Mary Rees and Tan Lei: they are expanding if and only if they do not admit a cycle of periodic rays.