# 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

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

## Citations in EuDML Documents

top- Veronica Baker, Marcy Barge, Jaroslaw Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts
- Clemens Fuchs, Robert Tijdeman, Substitutions, abstract number systems and the space filling property
- Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner, Dynamical directions in numeration

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