Growth series for expansion complexes

TL;DR

This paper investigates growth series for expansion complexes derived from finite subdivision rules, demonstrating polynomial growth and density of growth degrees in hyperbolic cases.

Contribution

It establishes that growth series for these complexes have polynomial growth and that hyperbolic expansion complexes exhibit a dense set of growth degrees.

Findings

Growth series have polynomial growth under specified norms.

Degrees of growth rates are dense in [2, ∞) for hyperbolic complexes.

Main theorem applies to complexes with bounded valence and mesh approaching zero.

Abstract

This paper is concerned with growth series for expansion complexes for finite subdivision rules. Suppose X is an expansion complex for a finite subdivision rule with bounded valence and mesh approaching 0, and let S be a seed for X. One can define a growth series for (X,S) by giving the tiles in the seed norm 0 and then using either the skinny path norm or the fat path norm to recursively define norms for the other tiles. The main theorem is that, with respect to either of these norms, the growth series for (X,S) has polynomial growth. Furthermore, the degrees of the growth rates of hyperbolic expansion complexes are dense in the ray [2,\infty).