Atomic surfaces, tilings and coincidences II. Reducible case

Hiromi Ei[1]; Shunji Ito[2]; Hui Rao[3]

  • [1] Chuo University Kasuga, Bunkyo-ku Department of Information and System Engineering Tokyo (Japan)
  • [2] Kanazawa University Department of Mathematical Kanazawa (Japan)
  • [3] Tsinghua University Department of Mathematics Beijing (China)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2285-2313
  • ISSN: 0373-0956

Abstract

top
The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that a non-periodic covering by the atomic surfaces covers the space exactly k -times.The atomic surfaces are originally designed by Rauzy to study the spectrum of the substitution dynamical system via a periodic tiling. However, we show that, since the stepped-surface is complicated in the reducible case, it is not clear whether the atomic surfaces can tile the space periodically or not. It seems that the geometry of the atomic surfaces can not applied directly to the spectral problem.

How to cite

top

Ei, Hiromi, Ito, Shunji, and Rao, Hui. "Atomic surfaces, tilings and coincidences II. Reducible case." Annales de l’institut Fourier 56.7 (2006): 2285-2313. <http://eudml.org/doc/10205>.

@article{Ei2006,
abstract = {The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that a non-periodic covering by the atomic surfaces covers the space exactly $k$-times.The atomic surfaces are originally designed by Rauzy to study the spectrum of the substitution dynamical system via a periodic tiling. However, we show that, since the stepped-surface is complicated in the reducible case, it is not clear whether the atomic surfaces can tile the space periodically or not. It seems that the geometry of the atomic surfaces can not applied directly to the spectral problem.},
affiliation = {Chuo University Kasuga, Bunkyo-ku Department of Information and System Engineering Tokyo (Japan); Kanazawa University Department of Mathematical Kanazawa (Japan); Tsinghua University Department of Mathematics Beijing (China)},
author = {Ei, Hiromi, Ito, Shunji, Rao, Hui},
journal = {Annales de l’institut Fourier},
keywords = {Atomic surfaces; Pisot substitution; tiling; atomic surfaces},
language = {eng},
number = {7},
pages = {2285-2313},
publisher = {Association des Annales de l’institut Fourier},
title = {Atomic surfaces, tilings and coincidences II. Reducible case},
url = {http://eudml.org/doc/10205},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Ei, Hiromi
AU - Ito, Shunji
AU - Rao, Hui
TI - Atomic surfaces, tilings and coincidences II. Reducible case
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2285
EP - 2313
AB - The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that a non-periodic covering by the atomic surfaces covers the space exactly $k$-times.The atomic surfaces are originally designed by Rauzy to study the spectrum of the substitution dynamical system via a periodic tiling. However, we show that, since the stepped-surface is complicated in the reducible case, it is not clear whether the atomic surfaces can tile the space periodically or not. It seems that the geometry of the atomic surfaces can not applied directly to the spectral problem.
LA - eng
KW - Atomic surfaces; Pisot substitution; tiling; atomic surfaces
UR - http://eudml.org/doc/10205
ER -

References

top
  1. S. Akiyama, Self-affine tiling and Pisot numeration system, Number theory and its applications (1999), 7-17, Kluwer Acad. Plubl., Dordrecht Zbl0999.11065MR1738803
  2. S. Akiyama, On the boundary of self-affine tilings generated by Pisot numbers, J. Math. Soc. Japan 54 (2002), 283-308 Zbl1032.11033MR1883519
  3. S. Akiyama, H. Rao, W. Steiner, A certain finiteness property of Pisot number systems, J. Number Theory 107 (2004), 135-160 Zbl1052.11055MR2059954
  4. P. Arnoux, V. Berthé, S. Ito, Discrete planes, ℤ 2 -actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble) 52 (2002), 305-349 Zbl1017.11006MR1906478
  5. P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. 8 (2001), 181-207 Zbl1007.37001MR1838930
  6. V. Baker, M. Barge, J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to β -shifts, (2005) Zbl1138.37008MR2180231
  7. C. Bandt, Self-similar sets. V. Integer matrices and fractal tilings of ℝ n , Proc. Amer. Math. Soc. 112 (1991), 549-562 Zbl0743.58027MR1036982
  8. M. Barge, B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France 130 (2002), 619-626 Zbl1028.37008MR1947456
  9. M. Barge, J. Kwapisz, Geometric theory of unimodular Pisot substitutions, (2004) Zbl1152.37011MR2262174
  10. J. Bernat, V. Berthé, H. Rao, On the super-coincidence condition, (2006) 
  11. V. Berthé, A. Siegel, Tilings associated with beta-numeration and substitutions, Electronic J. Comb. Number Theory 5 (2005) Zbl1139.37008MR2191748
  12. F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), 221-239 Zbl0348.54034MR461470
  13. F. Durand, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci. 65 (1989), 153-169 Zbl0679.10010MR1020484
  14. H. Ei, S. Ito, Tilings from some non-irreducible, Pisot substitutions, Discrete Mathematics and Theoretical Computer Science 7 (2005), 81-122 Zbl1153.37323MR2164061
  15. H. Ei, S. Ito, H. Rao, Atomic surfaces, tilings and coincidences III: β -tiling and super-coincidence 
  16. K. Falconer, Techniques in fractal geometry, (1997), John Wiley & Sons Ltd., Chichester Zbl0869.28003MR1449135
  17. N. Fogg, Substitutions in dynamics, arithmetics and combinatorics, (2002), Springer-Verlag, Berlin Zbl1014.11015MR1970385
  18. C. Frougny, B. Solomyak, Finite beta-expansions, Ergodic Theory & Dynam. Sys. 12 (1992), 713-723 Zbl0814.68065MR1200339
  19. B. Host 
  20. S. Ito, H. Rao, Atomic surfaces, tilings and coincidences I. Irreducible case Zbl1143.37013MR2254640
  21. S. Ito, H. Rao, Purely periodic β -expansions with Pisot unit base, Proc. Amer. Math. Soc. 133 (2005), 953-964 Zbl1099.11062MR2117194
  22. S. Ito, Y. Sano, On periodic β -expansions of Pisot numbers and Rauzy fractals, Osaka J. Math. 38 (2001), 349-368 Zbl0991.11040MR1833625
  23. R. Kenyon, Self-replicating tilings, 135 (1992), P. Walters, ed. Zbl0770.52013
  24. J. Lagarias, Y. Wang, Self-affine tiles in ℝ n , Adv. Math. 121 (1996), 21-49 Zbl0893.52013MR1399601
  25. J. Lagarias, Y. Wang, Substitution Delone sets, Discrete Comput. Geom. 29 (2003), 175-209 Zbl1037.52017MR1957227
  26. B. Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. 351 (1999), 3315-3349 Zbl0984.11008MR1615950
  27. M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, (1987), Springer-Verlag, Berlin Zbl0642.28013MR924156
  28. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178 Zbl0522.10032MR667748
  29. M. Senechal, Quasicrystals and Geometry, (1995), Cambridge University Press Zbl0828.52007MR1340198
  30. A. Siegel, Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot, (2000) 
  31. A. Siegel, Pure discrete spectrum dynamical system and periodic tiling associated with a substitution, Ann. Inst. Fourier (Grenoble) 54 (2004), 341-381 Zbl1083.37009MR2073838
  32. V. Sirvent, Y. Wang, Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math. 206 (2002), 465-485 Zbl1048.37015MR1926787
  33. W. P. Thurston, Groups, tilings, and finite state automata, (1989), Providence, RI 
  34. J. Thuswaldner, Unimodular Pisot substitutions and their associated tiles Zbl1161.37016
  35. A. Vince, Digit tiling of Euclidean space, Center de Recherches Mathematiques CRM Monograph Series 13 (2000), 329-370 Zbl0972.52012MR1798999

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.