Expansion properties for finite subdivision rules I

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.

Levy and Thurston obstructions of finite subdivision rules

“…The case A " P f of the following theorem is in [FPP18b], and the same proof works for this generalization. Theorem 6.8 ([FPP18b, Theorem 6.5]).…”

“…Backward contracting finite subdivision rules. We refer the reader to [FPP18b] for details of this section.…”

Expansion properties for finite subdivision rules II

“…This paper continues our study of expansion properties for finite subdivision rules begun in [12]. In [12] we usually began with a Thurston map which is the subdivision map of a finite subdivision rule. Here we begin with a general Thurston map, and we ask whether it is isotopic to the subdivision map of a finite subdivision rule.…”

“…This paper continues our study of expansion properties for finite subdivision rules begun in [12]. In [12] we usually began with a Thurston map which is the subdivision map of a finite subdivision rule.…”

“…If a Thurston map f is the subdivision map of a finite subdivision rule, then the lift of f −1 to the universal covering orbifold of f is combinatorially distance nonincreasing. This is formally stated in statement 1 of Lemma 6.1 of [12]. We use this result in this section, and in the next section we use its generalization to the universal covering space of the complement in S 2 of the postcritical set of f .…”