Expansion properties for finite subdivision rules I

TL;DR

This paper explores the expansion properties of Thurston maps that are derived from finite subdivision rules, examining their combinatorial, dynamical, algebraic, and coarse-geometric aspects.

Contribution

It establishes connections between different notions of expansion for Thurston maps arising from finite subdivision rules.

Findings

Identifies relationships between combinatorial and dynamical expansion.

Analyzes algebraic and coarse-geometric expansion properties.

Provides a framework for understanding expansion in subdivision maps.

Abstract

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion---combinatorial, dynamical, algebraic, and coarse-geometric.