Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Generalized $\beta$-expansions, substitution tilings, and local finiteness
HTML articles powered by AMS MathViewer

by Natalie Priebe Frank and E. Arthur Robinson Jr. PDF
Trans. Amer. Math. Soc. 360 (2008), 1163-1177 Request permission


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.
  • L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Tr. Mat. Inst. Steklova 239 (2002), no. Diskret. Geom. i Geom. Chisel, 118–126; English transl., Proc. Steklov Inst. Math. 4(239) (2002), 108–116. MR 1975139
  • N. P. Frank, Non-constant length ${\mathbb {Z}}^d$ substitutions, in preparation.
  • D. Frettlöh, Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor, Ph.D. dissertation, University of Dortmund, 2002.
  • Richard Kenyon, Rigidity of planar tilings, Invent. Math. 107 (1992), no. 3, 637–651. MR 1150605, DOI 10.1007/BF01231905
  • A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
  • E. Arthur Robinson Jr., Symbolic dynamics and tilings of $\Bbb R^d$, Symbolic dynamics and its applications, Proc. Sympos. Appl. Math., vol. 60, Amer. Math. Soc., Providence, RI, 2004, pp. 81–119. MR 2078847, DOI 10.1090/psapm/060/2078847
  • L. Sadun, Some generalizations of the pinwheel tiling, Discrete Comput. Geom. 20 (1998), no. 1, 79–110. MR 1626703, DOI 10.1007/PL00009379
  • Klaus Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269–278. MR 576976, DOI 10.1112/blms/12.4.269
  • Boris Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695–738. MR 1452190, DOI 10.1017/S0143385797084988
  • W. Thurston, Groups, Tilings, and Finite State Automata, AMS Colloquium Lecture Notes, American Mathematical Society, Boulder, 1989.
  • Keith M. Wilkinson, Ergodic properties of a class of piecewise linear transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 303–328. MR 374390, DOI 10.1007/BF00532869
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52C20, 37B50
  • Retrieve articles in all journals with MSC (2000): 52C20, 37B50
Additional Information
  • Natalie Priebe Frank
  • Affiliation: Department of Mathematics, Vassar College, Box 248, Poughkeepsie, New York 12604
  • Email:
  • E. Arthur Robinson Jr.
  • Affiliation: Department of Mathematics, George Washington University, Washington, DC 20052
  • Email:
  • Received by editor(s): June 6, 2005
  • Published electronically: October 23, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1163-1177
  • MSC (2000): Primary 52C20; Secondary 37B50
  • DOI:
  • MathSciNet review: 2357692