Generalized $\beta$-expansions, substitution tilings, and local finiteness
Frank, Natalie Priebe ; Robinson, Jr., E. Arthur
arXiv, 0506098 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2005-06-06
Classification:  Mathematics - Dynamical Systems,  Mathematical Physics,  Mathematics - Metric Geometry,  Primary 52C20,  Secondary 37B50 
@article{0506098,
     author = {Frank, Natalie Priebe and Robinson, Jr., E. Arthur},
     title = {Generalized $\beta$-expansions, substitution tilings, and local
  finiteness},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0506098}
}

Frank, Natalie Priebe; Robinson, Jr., E. Arthur. Generalized $\beta$-expansions, substitution tilings, and local
  finiteness. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0506098/