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

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

2005-06-06

arxiv.org/abs/math/0506098v2

Mathematics

Dynamical Systems

14 pages, 7 figures. Accepted for publication in the Transactions of the AMS. Reference added, typos fixed, and minor edits re

Scientific paper

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $\beta$ 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 $\beta$, 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 $\beta$-transformation.

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