Substitutions for tilings \p,q\

2006/11/09 by Margenstern, Maurice, Skordev, Guentcho
#Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #G.2

paper · doi:10.48550/arxiv.cs/0611039

Abstract

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

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