Transactions of the American Mathematical Society

Author(s): Natalie Priebe Frank; E. Arthur Robinson Jr.
Journal: Trans. Amer. Math. Soc. 360 (2008), 1163-1177.
MSC (2000): Primary 52C20; Secondary 37B50.
Posted: October 23, 2007
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Abstract: 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.

1.: L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math. 239 (2002), pp. 118-126. MR 1975139 (2004c:52033)
2.: N. P. Frank, Non-constant length ${\mathbb{Z}}^d$ substitutions, in preparation.
3.: D. Frettlöh, Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor, Ph.D. dissertation, University of Dortmund, 2002.
4.: R. Kenyon, Rigidity of planar tilings, Invent. Math. 107 (1992), 637-651. MR 1150605 (92m:52049)
5.: A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477-493. MR 0097374 (20:3843)
6.: E. A. Robinson, Symbolic dynamics and tilings of ${\mathbb{R}}^d$ , Proceedings of Symposia in Applied Mathematics 60 (2004), 81-119. MR 2078847 (2005h:37036)
7.: L. Sadun, Some generalizations of the pinwheel tiling, Discrete Comp. Geom. 20, no. 1 (1998), 79-110. MR 1626703 (99e:52029)
8.: K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269-278. MR 576976 (82c:12003)
9.: B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynamical Systems 17 (1997), 695-738. Errata, Ergodic Theory Dynamical Systems 19 (1999), 1685. MR 1452190 (98f:52030)
10.: W. Thurston, Groups, Tilings, and Finite State Automata, AMS Colloquium Lecture Notes, American Mathematical Society, Boulder, 1989.
11.: K. M. Wilkinson, Ergodic properties of a class of piecewise linear transformations, Z. Wahr. verw. Gebiete 31 (1975), pp. 303-328. MR 0374390 (51:10590)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52C20, 37B50.

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: 10.1090/S0002-9947-07-04527-8
PII: S 0002-9947(07)04527-8
Keywords: Substitution sequence, self-similar tiling
Received by editor(s): June 6, 2005
Posted: October 23, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

Alex Clark, The dynamics of tiling spaces, Open Problems in Topology II, Elsevier, 2007, pp. 463-467.

Elliott Pearl, The dynamics of tiling spaces, Open Problems in Topology II, Elsevier, 2007, pp. 463-467.

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