Substitutions for tilings $\{p,q\}$

TL;DR

This paper explores the combinatorial structure of tilings {p,q} in Euclidean and hyperbolic spaces, introducing substitutions and recurrence relations to analyze tile growth and graph properties.

Contribution

It constructs a substitution { sigma_{q,p}} for tilings { p,q } and links the tile count growth to the dominant root of a characteristic polynomial.

Findings

The substitution { sigma_{q,p}} generates a spanning tree of the tiling graph.

The tile count sequence {u_n} satisfies a linear recurrence.

Growth rate of tiles is determined by the dominant root of the recurrence's characteristic polynomial.

Abstract

In this paper we consider tiling $\{p, q \}$ of the Euclidean space and of the hyperbolic space, and its dual graph $\Gamma_{q, p}$ from a combinatorial point of view. A substitution $\sigma_{q, p}$ on an appropriate finite alphabet is constructed. The homogeneity of graph $\Gamma_{q, p}$ and its generation function are the basic tools for the construction. The tree associated with substitution $\sigma_{q, p}$ is a spanning tree of graph $\Gamma_{q, p}$. Let $u_n$ be the number of tiles of tiling $\{p, q \}$ of generation $n$. The characteristic polynomial of the transition matrix of substitution $\sigma_{q, p}$ is a characteristic polynomial of a linear recurrence. The sequence $(u_n)_{n \geq 0}$ is a solution of this recurrence. The growth of sequence $(u_n)_{n \geq 0}$ is given by the dominant root of the characteristic polynomial.