Expansion properties for finite subdivision rules II

TL;DR

This paper characterizes when iterates of Thurston maps can be represented as subdivision maps of finite subdivision rules, providing criteria based on their covering properties and Levy cycles.

Contribution

It proves that large iterates of certain Thurston maps are isotopic to finite subdivision rule maps and identifies conditions preventing this.

Findings

Large iterates of non-torus-covered Thurston maps without Levy cycles are isotopic to subdivision maps.

Characterization of Thurston maps doubly covered by torus endomorphisms with such isotopies.

Conditions under which no iterate of a Thurston map is isotopic to a subdivision map.

Abstract

We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.