2005/06/06 by Natalie Priebe Frank, Frank, Natalie Priebe, Jr. Robinson +2
Computer Science · Materials Science · Mathematics · Physics and Astronomy · Social Sciences · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Gender Studies in Language #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Primary 52C20 #Quasicrystal Structures and Properties #Secondary 37B50 #math-ph #math.DS #math.MG #math.MP #msc:37B50 #msc:52C20 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0506098
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
openalex publication_date 2005/06/06 · arxiv created 2005/11/08 · arxiv updated 2012/08/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion β 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, 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-transformation.