## Generalized $\beta$-expansions, substitution tilings, and local finiteness

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

## 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.## References

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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
- 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: https://doi.org/10.1090/S0002-9947-07-04527-8
- MathSciNet review: 2357692