St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Parametrization of a two-dimensional quasiperiodic Rauzy tiling

Author: V. G. Zhuravlev
Translated by: B. Bekker
Original publication: Algebra i Analiz, tom 22 (2010), nomer 4.
Journal: St. Petersburg Math. J. 22 (2011), 529-555
MSC (2010): Primary 05B45, 52C20, 37B50
Published electronically: May 2, 2011
MathSciNet review: 2768960
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Abstract: With the help of an affine inflation $ B$, a two-dimensional quasiperiodic Rauzy tiling $ \mathcal{R}^\infty$ is constructed, together with a parametrization of its tiles by algebraic integers $ \mathbb{Z}[\zeta] \subset [0,1)$, where $ \zeta$ is a certain Pisot number (specifically, the real root of the polynomial $ x^3+x^2+x-1$). The coronas (clusters) of the tiling $ \mathcal{R}^\infty$ are classified by disjoint half-intervals in $ [0,1)$ the lengths of which are proportional to the frequencies of the corresponding corona types. It is proved that, for each of the three basis tiles, there exists an odd number of corona types of an arbitrary level. The parametrization obtained describes local rules (tile adjacency conditions) for $ \mathcal{R}^\infty$, and it conjugates the action of the affine rotation $ B$ of the plane $ \mathbb{R}^2$ by an irrational angle with a shift of the coding sequences.

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Additional Information

V. G. Zhuravlev
Affiliation: Vladimir State Pedagogical University, Stroitelei 11, Vladimir 600024, Russia

Keywords: Quasiperiodic tilings, local rules, divisible figures, two-dimensional approximations
Received by editor(s): April 1, 2009
Published electronically: May 2, 2011
Additional Notes: Supported by RFBR (grant no. 08-01-00326)
Article copyright: © Copyright 2011 American Mathematical Society