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Generalized $β$-expansions, substitution tilings, and local finiteness

Natalie Priebe Frank·E. Arthur Robinson, Jr.
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Abstract

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $β$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $β$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $β$-transformation.

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BibTeXRIS

Natalie Priebe Frank, E. Arthur Robinson, Jr.. 2005-11-08. Generalized $β$-expansions, substitution tilings, and local finiteness. https://arxiv.org/abs/math/0506098

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