Generalized -expansions, substitution tilings, and local finiteness

Authors:
Natalie Priebe Frank and E. Arthur Robinson Jr.

Journal:
Trans. Amer. Math. Soc. **360** (2008), 1163-1177

MSC (2000):
Primary 52C20; Secondary 37B50.

DOI:
https://doi.org/10.1090/S0002-9947-07-04527-8

Published electronically:
October 23, 2007

MathSciNet review:
2357692

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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**Natalie Priebe Frank**

Affiliation:
Department of Mathematics, Vassar College, Box 248, Poughkeepsie, New York 12604

Email:
nafrank@vassar.edu

**E. Arthur Robinson Jr.**

Affiliation:
Department of Mathematics, George Washington University, Washington, DC 20052

Email:
robinson@gwu.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04527-8

Keywords:
Substitution sequence,
self-similar tiling

Received by editor(s):
June 6, 2005

Published electronically:
October 23, 2007

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.