Generalized 𝛽-expansions, substitution tilings, and local finiteness
Natalie Priebe Frank, E. Arthur Robinson · Transactions of the American Mathematical Society · 2007
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion β \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 β \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 β \beta -transformation.