Generalized $\beta$-expansions, substitution tilings, and local finiteness
Dynamical Systems
2012-08-27 v2 Mathematical Physics
Metric Geometry
math.MP
Abstract
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion 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 , 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 -transformation.
Cite
@article{arxiv.math/0506098,
title = {Generalized $\beta$-expansions, substitution tilings, and local finiteness},
author = {Natalie Priebe Frank and E. Arthur Robinson,},
journal= {arXiv preprint arXiv:math/0506098},
year = {2012}
}
Comments
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