Remote Access St. Petersburg Mathematical Journal

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. V. G. Zhuravlev, Rauzy tilings and bounded remainder sets on a torus, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 322 (2005), no. Trudy po Teorii Chisel, 83–106, 253 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 137 (2006), no. 2, 4658–4672. MR 2138453,
  • 2. V. G. Zhuravlev, One-dimensional Fibonacci tilings, Izv. Ross. Akad. Nauk Ser. Mat. 71 (2007), no. 2, 89–122 (Russian, with Russian summary); English transl., Izv. Math. 71 (2007), no. 2, 307–340. MR 2316983,
  • 3. V. G. Zhuravlev, On additivity property of the complexity function related to Rauzy tiling, Analytic and probabilistic methods in number theory/Analiziniai ir tikimybiniai metodai skaičių teorijoje, TEV, Vilnius, 2007, pp. 240–254. MR 2397156
  • 4. V. G. Zhuravlev and A. V. Maleev, The complexity function and forsing in two-dimensional quasiperiodic Rauzy tiling, Kristallografiya 52 (2007), no. 4, 610-616. (Russian)
  • 5. -, The layer-by-layer growth of a quasiperiodic Rauzy tiling, Kristallografiya 52 (2007), no. 2, 204-210. (Russian)
  • 6. -, Quasiperiods of layer-by-layer growth of the Rauzy tiling, Kristallografiya 52 (2007), no. 1, 7-14. (Russian)
  • 7. -, The diffraction of the two-dimensional quasiperiodic Rauzy tiling, Kristallografiya 53 (2008), no. 6, 978-986. (Russian)
  • 8. -, Construction of the two-dimensional quasiperiodic Rauzy tiling by means of similarity transformation, Kristallografiya 54 (2009), no. 3, 389-399. (Russian)
  • 9. A. V. Maleev, A. V. Shutov, and V. G. Zhuravlev, Quasiperiodic tilings of plane constructed on the basis of cubic irrationalities, 27th N. V. Belov Scientific Readings, Nizhni Novgorod, 2008, pp. 29-31. (Russian)
  • 10. Shigeki Akiyama, Self affine tiling and Pisot numeration system, Number theory and its applications (Kyoto, 1997) Dev. Math., vol. 2, Kluwer Acad. Publ., Dordrecht, 1999, pp. 7–17. MR 1738803
  • 11. Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998) de Gruyter, Berlin, 2000, pp. 11–26. MR 1770451
  • 12. P. Arnoux, V. Berthé, and S. Ito, Discrete planes, ℤ²-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 305–349 (English, with English and French summaries). MR 1906478
  • 13. N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. MR 1970385
  • 14. Shunji Ito and Makoto Ohtsuki, Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math. 16 (1993), no. 2, 441–472. MR 1247666,
  • 15. Shunji Ito, Junko Fujii, Hiroko Higashino, and Shin-ichi Yasutomi, On simultaneous approximation to (𝛼,𝛼²) with 𝛼³+𝑘𝛼-1=0, J. Number Theory 99 (2003), no. 2, 255–283. MR 1968452,
  • 16. Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1972. North-Holland Mathematical Library, Vol. 2. MR 0485769
  • 17. Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403–441. MR 1460032
  • 18. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178. MR 0667748 (84h:10074)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 05B45, 52C20, 37B50

Retrieve articles in all journals with MSC (2010): 05B45, 52C20, 37B50

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

American Mathematical Society