Generalized $\beta$expansions, substitution tilings, and local finiteness
Abstract
For a fairly general class of twodimensional 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 twodimensional substitution on rectangular tiles, with a nonPisot length expansion $\beta$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively onedimensional and involve the idea of a certain type of generalized $\beta$transformation.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2005
 arXiv:
 arXiv:math/0506098
 Bibcode:
 2005math......6098P
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Metric Geometry;
 Mathematics  Mathematical Physics;
 Mathematical Physics;
 Primary 52C20;
 Secondary 37B50
 EPrint:
 14 pages, 7 figures. Accepted for publication in the Transactions of the AMS. Reference added, typos fixed, and minor edits reflecting the referee's report