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Hiromi Ei; Shunji Ito; Hui Rao
Atomic surfaces, tilings and coincidences II. Reducible case
(Surfaces atomiques, pavages et coïncidences II)
Annales de l'institut Fourier, 56 no. 7 (2006), p. 2285-2313, doi: 10.5802/aif.2241
Article PDF | Reviews MR 2290782 | Zbl 1119.52013 | 3 citations in Cedram
Class. Math.: 52C23, 37A45, 28A80, 11B85
Keywords: Atomic surfaces, Pisot substitution, tiling
[1] S. Akiyama, Self-affine tiling and Pisot numeration system, Number theory and its applications, Dev. Math., Kluwer Acad. Plubl., 1999, p. 7-17 MR 1738803 | Zbl 0999.11065
[2] S. Akiyama, “On the boundary of self-affine tilings generated by Pisot numbers”, J. Math. Soc. Japan 54 (2002) no. 2, p. 283-308 Article | MR 1883519 | Zbl 1032.11033
[3] S. Akiyama, H. Rao & W. Steiner, “A certain finiteness property of Pisot number systems”, J. Number Theory 107 (2004), p. 135-160 Article | MR 2059954 | Zbl 1052.11055
[4] P. Arnoux, V. Berthé & S. Ito, “Discrete planes, ${\mathbb{Z}}^2$-actions, Jacobi-Perron algorithm and substitutions”, Ann. Inst. Fourier (Grenoble) 52 (2002) no. 2, p. 305-349 Cedram | MR 1906478 | Zbl 1017.11006
[5] P. Arnoux & S. Ito, “Pisot substitutions and Rauzy fractals”, Bull. Belg. Math. Soc. 8 (2001), p. 181-207 Article | MR 1838930 | Zbl 1007.37001
[6] V. Baker, M. Barge & J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts, 2005 MR 2180231
[7] C. Bandt, “Self-similar sets. V. Integer matrices and fractal tilings of ${\mathbb{R}}^ n$”, Proc. Amer. Math. Soc. 112 (1991) no. 2, p. 549-562 MR 1036982 | Zbl 0743.58027
[8] M. Barge & B. Diamond, “Coincidence for substitutions of Pisot type”, Bull. Soc. Math. France 130 (2002) no. 4, p. 619-626 Numdam | MR 1947456 | Zbl 1028.37008
[9] M. Barge & J. Kwapisz, “Geometric theory of unimodular Pisot substitutions”, Preprint, 2004 MR 2262174 | Zbl 05071304
[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) no. 3 MR 2191748 | Zbl 05014493
[12] F. M. Dekking, “The spectrum of dynamical systems arising from substitutions of constant length”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78) no. 3, p. 221-239 Article | MR 461470 | Zbl 0348.54034
[13] F. Durand & A. Thomas, “Systèmes de numération et fonctions fractales relatifs aux substitutions”, Theoret. Comput. Sci. 65 (1989) no. 2, p. 153-169 Article | MR 1020484 | Zbl 0679.10010
[14] H. Ei & S. Ito, “Tilings from some non-irreducible, Pisot substitutions”, Discrete Mathematics and Theoretical Computer Science 7 (2005) no. 1, p. 81-122 MR 2164061 | Zbl 1153.37323
[15] H. Ei, S. Ito & H. Rao, “Atomic surfaces, tilings and coincidences III: $\beta $-tiling and super-coincidence”, In preparation
[16] K. Falconer, Techniques in fractal geometry, John Wiley & Sons Ltd., Chichester, 1997 MR 1449135 | Zbl 0869.28003
[17] N. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002 MR 1970385 | Zbl 1014.11015
[18] C. Frougny & B. Solomyak, “Finite beta-expansions”, Ergodic Theory & Dynam. Sys. 12 (1992), p. 713-723 MR 1200339 | Zbl 0814.68065
[19] B. Host, unpublished manuscript
[20] S. Ito & H. Rao, “Atomic surfaces, tilings and coincidences I. Irreducible case”, To appear in Israel J. Math. MR 2254640 | Zbl 1143.37013
[21] S. Ito & H. Rao, “Purely periodic $\beta $-expansions with Pisot unit base”, Proc. Amer. Math. Soc. 133 (2005), p. 953-964 Article | MR 2117194 | Zbl 02125243
[22] S. Ito & Y. Sano, “On periodic $\beta $-expansions of Pisot numbers and Rauzy fractals”, Osaka J. Math. 38 (2001), p. 349-368 Article | MR 1833625 | Zbl 0991.11040
[23] R. Kenyon, Self-replicating tilings, Symbolic dynamics and its applications, Contemporary mathematics series 135, P. Walters, ed., 1992 Zbl 0770.52013
[24] J. Lagarias & Y. Wang, “Self-affine tiles in ${\mathbb{R}}^ n$”, Adv. Math. 121 (1996) no. 1, p. 21-49 Article | MR 1399601 | Zbl 0893.52013
[25] J. Lagarias & Y. Wang, “Substitution Delone sets”, Discrete Comput. Geom. 29 (2003) no. 2, p. 175-209 MR 1957227 | Zbl 1037.52017
[26] B. Praggastis, “Numeration systems and Markov partitions from self-similar tilings”, Trans. Amer. Math. Soc. 351 (1999) no. 8, p. 3315-3349 Article | MR 1615950 | Zbl 0984.11008
[27] M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math., 1294, Springer-Verlag, Berlin, 1987 MR 924156 | Zbl 0642.28013
[28] G. Rauzy, “Nombres algébriques et substitutions”, Bull. Soc. Math. France 110 (1982), p. 147-178 Numdam | MR 667748 | Zbl 0522.10032
[29] M. Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995 MR 1340198 | Zbl 0828.52007
[30] A. Siegel, Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot, Thèse de doctorat, Université de la Méditérranée, 2000
[31] A. Siegel, “Pure discrete spectrum dynamical system and periodic tiling associated with a substitution”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 2, p. 341-381 Cedram | MR 2073838 | Zbl 1083.37009
[32] V. Sirvent & Y. Wang, “Self-affine tiling via substitution dynamical systems and Rauzy fractals”, Pacific J. Math. 206 (2002) no. 2, p. 465-485 Article | MR 1926787 | Zbl 1048.37015
[33] W. P. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lectures, Providence, RI, 1989
[34] J. Thuswaldner, “Unimodular Pisot substitutions and their associated tiles”, To appear in J. Théor. Nombres Bordeaux Cedram | Zbl 05135401
[35] A. Vince, “Digit tiling of Euclidean space”, Center de Recherches Mathematiques CRM Monograph Series 13 (2000), p. 329-370 MR 1798999 | Zbl 0972.52012
Hiromi Ei; Shunji Ito; Hui Rao
Atomic surfaces, tilings and coincidences II. Reducible case
(Surfaces atomiques, pavages et coïncidences II)
Annales de l'institut Fourier, 56 no. 7 (2006), p. 2285-2313, doi: 10.5802/aif.2241
Article PDF | Reviews MR 2290782 | Zbl 1119.52013 | 3 citations in Cedram
Class. Math.: 52C23, 37A45, 28A80, 11B85
Keywords: Atomic surfaces, Pisot substitution, tiling
Résumé - 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.
Bibliography
[2] S. Akiyama, “On the boundary of self-affine tilings generated by Pisot numbers”, J. Math. Soc. Japan 54 (2002) no. 2, p. 283-308 Article | MR 1883519 | Zbl 1032.11033
[3] S. Akiyama, H. Rao & W. Steiner, “A certain finiteness property of Pisot number systems”, J. Number Theory 107 (2004), p. 135-160 Article | MR 2059954 | Zbl 1052.11055
[4] P. Arnoux, V. Berthé & S. Ito, “Discrete planes, ${\mathbb{Z}}^2$-actions, Jacobi-Perron algorithm and substitutions”, Ann. Inst. Fourier (Grenoble) 52 (2002) no. 2, p. 305-349 Cedram | MR 1906478 | Zbl 1017.11006
[5] P. Arnoux & S. Ito, “Pisot substitutions and Rauzy fractals”, Bull. Belg. Math. Soc. 8 (2001), p. 181-207 Article | MR 1838930 | Zbl 1007.37001
[6] V. Baker, M. Barge & J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts, 2005 MR 2180231
[7] C. Bandt, “Self-similar sets. V. Integer matrices and fractal tilings of ${\mathbb{R}}^ n$”, Proc. Amer. Math. Soc. 112 (1991) no. 2, p. 549-562 MR 1036982 | Zbl 0743.58027
[8] M. Barge & B. Diamond, “Coincidence for substitutions of Pisot type”, Bull. Soc. Math. France 130 (2002) no. 4, p. 619-626 Numdam | MR 1947456 | Zbl 1028.37008
[9] M. Barge & J. Kwapisz, “Geometric theory of unimodular Pisot substitutions”, Preprint, 2004 MR 2262174 | Zbl 05071304
[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) no. 3 MR 2191748 | Zbl 05014493
[12] F. M. Dekking, “The spectrum of dynamical systems arising from substitutions of constant length”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78) no. 3, p. 221-239 Article | MR 461470 | Zbl 0348.54034
[13] F. Durand & A. Thomas, “Systèmes de numération et fonctions fractales relatifs aux substitutions”, Theoret. Comput. Sci. 65 (1989) no. 2, p. 153-169 Article | MR 1020484 | Zbl 0679.10010
[14] H. Ei & S. Ito, “Tilings from some non-irreducible, Pisot substitutions”, Discrete Mathematics and Theoretical Computer Science 7 (2005) no. 1, p. 81-122 MR 2164061 | Zbl 1153.37323
[15] H. Ei, S. Ito & H. Rao, “Atomic surfaces, tilings and coincidences III: $\beta $-tiling and super-coincidence”, In preparation
[16] K. Falconer, Techniques in fractal geometry, John Wiley & Sons Ltd., Chichester, 1997 MR 1449135 | Zbl 0869.28003
[17] N. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002 MR 1970385 | Zbl 1014.11015
[18] C. Frougny & B. Solomyak, “Finite beta-expansions”, Ergodic Theory & Dynam. Sys. 12 (1992), p. 713-723 MR 1200339 | Zbl 0814.68065
[19] B. Host, unpublished manuscript
[20] S. Ito & H. Rao, “Atomic surfaces, tilings and coincidences I. Irreducible case”, To appear in Israel J. Math. MR 2254640 | Zbl 1143.37013
[21] S. Ito & H. Rao, “Purely periodic $\beta $-expansions with Pisot unit base”, Proc. Amer. Math. Soc. 133 (2005), p. 953-964 Article | MR 2117194 | Zbl 02125243
[22] S. Ito & Y. Sano, “On periodic $\beta $-expansions of Pisot numbers and Rauzy fractals”, Osaka J. Math. 38 (2001), p. 349-368 Article | MR 1833625 | Zbl 0991.11040
[23] R. Kenyon, Self-replicating tilings, Symbolic dynamics and its applications, Contemporary mathematics series 135, P. Walters, ed., 1992 Zbl 0770.52013
[24] J. Lagarias & Y. Wang, “Self-affine tiles in ${\mathbb{R}}^ n$”, Adv. Math. 121 (1996) no. 1, p. 21-49 Article | MR 1399601 | Zbl 0893.52013
[25] J. Lagarias & Y. Wang, “Substitution Delone sets”, Discrete Comput. Geom. 29 (2003) no. 2, p. 175-209 MR 1957227 | Zbl 1037.52017
[26] B. Praggastis, “Numeration systems and Markov partitions from self-similar tilings”, Trans. Amer. Math. Soc. 351 (1999) no. 8, p. 3315-3349 Article | MR 1615950 | Zbl 0984.11008
[27] M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math., 1294, Springer-Verlag, Berlin, 1987 MR 924156 | Zbl 0642.28013
[28] G. Rauzy, “Nombres algébriques et substitutions”, Bull. Soc. Math. France 110 (1982), p. 147-178 Numdam | MR 667748 | Zbl 0522.10032
[29] M. Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995 MR 1340198 | Zbl 0828.52007
[30] A. Siegel, Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot, Thèse de doctorat, Université de la Méditérranée, 2000
[31] A. Siegel, “Pure discrete spectrum dynamical system and periodic tiling associated with a substitution”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 2, p. 341-381 Cedram | MR 2073838 | Zbl 1083.37009
[32] V. Sirvent & Y. Wang, “Self-affine tiling via substitution dynamical systems and Rauzy fractals”, Pacific J. Math. 206 (2002) no. 2, p. 465-485 Article | MR 1926787 | Zbl 1048.37015
[33] W. P. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lectures, Providence, RI, 1989
[34] J. Thuswaldner, “Unimodular Pisot substitutions and their associated tiles”, To appear in J. Théor. Nombres Bordeaux Cedram | Zbl 05135401
[35] A. Vince, “Digit tiling of Euclidean space”, Center de Recherches Mathematiques CRM Monograph Series 13 (2000), p. 329-370 MR 1798999 | Zbl 0972.52012
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