Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to β-shifts
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, p. 2213-2248
This article is devoted to the study of the translation flow on self-similar tilings associated with a substitution of Pisot type. We construct a geometric representation and give necessary and sufficient conditions for the flow to have pure discrete spectrum. As an application we demonstrate that, for certain beta-shifts, the natural extension is naturally isomorphic to a toral automorphism.
Cet article est consacré à l’étude du flot de translation sur pavages auto-similaires associés à une substitution de type Pisot. Nous construisons une représentation géométrique et nous donnons les conditions nécessaires et suffisantes pour que le flot ait un spectre purement discret. Dans l’application, nous montrons que pour certains beta-shifts, l’extension naturelle est naturellement isomorphique à un automorphisme du tore.
DOI : https://doi.org/10.5802/aif.2238
Classification:  37B50,  11R06,  28D05
Keywords: Substitution, tilings, pure discrete spectrum spectrum, Pisot
@article{AIF_2006__56_7_2213_0,
     author = {Baker, Veronica and Barge, Marcy and Kwapisz, Jaroslaw},
     title = {Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     pages = {2213-2248},
     doi = {10.5802/aif.2238},
     zbl = {1138.37008},
     mrnumber = {2290779},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2006__56_7_2213_0}
}
Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2213-2248. doi : 10.5802/aif.2238. https://aif.centre-mersenne.org/item/AIF_2006__56_7_2213_0/

[1] Akiyama, S. On the boundary of self affine tilings generated by Pisot numbers, J. Math. Soc. Japan, Tome 54 (2002) no. 2, pp. 283-308 | Article | MR 1883519 | Zbl 1032.11033

[2] Akiyama, S.; Rao, H.; Steiner, W. A certain finiteness property of Pisot number systems, J. Number Theory, Tome 107 (2004) no. 1, pp. 135-160 | Article | MR 2059954 | Zbl 1052.11055

[3] Arnoux, P.; Ito, S. Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Tome 8 (2001) no. 2, pp. 181-207 (Journées Montoises (Marne-la-Vallée, 2000)) | MR 1838930 | Zbl 1007.37001

[4] Barge, M.; Diamond, B. A complete invariant for the topology of one-dimensional substitution tiling spaces., Ergodic Theory Dynam. Systems, Tome 21 (2001) no. 5, pp. 1333-1358 | Article | MR 1855835 | Zbl 0986.37015

[5] Barge, M.; Kwapisz, J. Geometric Theory of Unimodular Pisot Substitutions, American J. of Math., Tome 128 (2006), pp. 1219-1282 | Article | MR 2262174 | Zbl 05071304

[6] Barge, M.; Kwapisz, J. Elements of the theory of unimodular Pisot substitutions with an application to β-shifts, Algebraic and Topological Dynamics, Amer. Math. Soc., Providence, RI (Contemporary Mathematics, Volume: 385) (Nov 2005), pp. 89-99 | MR 2180231 | Zbl 02236661

[7] Berthé, V.; Siegel, A. Tilings associated with beta-numeration and substitutions, Integers: Electronic journal of Combinatorial Number Theory, Tome 5 (2005) no. 3, pp. A02 | MR 2191748 | Zbl 05014493

[8] Bertrand, A. Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris, Tome 285 (1977) no. 6, p. A419-A421 | MR 447134 | Zbl 0362.10040

[9] Canterini, V.; Siegel, A. Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc., Tome 353 (2001) no. 12, pp. 5121-5144 | Article | MR 1852097 | Zbl 01663181

[10] Clark, A.; Sadun, L. When size matters: subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems, Tome 23 (2003), pp. 1043-1057 | Article | MR 1997967 | Zbl 1042.37008

[11] Ei, H.; Ito, S. Tilings from some non-irreducible, Pisot substitutions, Discrete Math. and Theo. Comp. Science, Tome 8 (2005) no. 1, pp. 81-122 | MR 2164061 | Zbl 1153.37323

[12] Ei, H.; Ito, S.; Rao, H. Atomic surfaces, tilings and coincidences II: Reducible case. (2006) (to appear in Annal. Institut Fourier (Grenoble)) | Numdam

[13] Frougny, C.; Solomyak, B. Finite beta-expansions, Ergodic Theory Dynam. Systems, Tome 12 (1992) no. 4, pp. 713-723 | Article | MR 1200339 | Zbl 0814.68065

[14] Hollander, M. Linear Numeration Systems, Finite Beta Expansions, and Discrete Spectrum of Substitution Dynamical Systems, University of Washington (1996) (Ph. D. Thesis)

[15] Host, B. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable, Ergodic Theory Dynam. Systems, Tome 6 (1986) no. 4, pp. 529-540 | Article | MR 873430 | Zbl 0625.28011

[16] Ito, S.; Rao, H. Atomic surfaces, tilings and coincidences I: Irreducible case, Isreal J. of Math., Tome 153 (2006), pp. 129-156 | Article | MR 2254640 | Zbl 1143.37013

[17] Kenyon, R.; Vershik, A. Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems, Tome 18 (1998) no. 2, pp. 357-372 | Article | MR 1619562 | Zbl 0915.58077

[18] Kwapisz, J. Dynamical Proof of Pisot’s Theorem, Canad. Math. Bull., Tome 49 (2006) no. 1, pp. 108-112 | Article | MR 2198723 | Zbl 05038781

[19] Mossé, B. Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci., Tome 99 (1992) no. 2, pp. 327-334 | Article | MR 1168468 | Zbl 0763.68049

[20] Queffélec, M. Substitution dynamical systems-spectral analysis, Springer-Verlag, Berlin (1987) (Lecture Notes in Mathematics, Vol. 1294) | MR 924156 | Zbl 0642.28013

[21] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Tome 110 (1982) no. 2, pp. 147-178 | Numdam | MR 667748 | Zbl 0522.10032

[22] Schmidt, K. On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., Tome 12 (1980), pp. 269-278 | Article | MR 576976 | Zbl 0494.10040

[23] Schmidt, K. Algebraic coding of expansive group automorphisms and two-sided beta-shifts, Mh. Math., Tome 129 (2000), pp. 37-61 | Article | MR 1741033 | Zbl 1010.37005

[24] Sidorov, N. Bijective and general arithmetic codings for Pisot toral automorphisms, J. Dynam. Control Systems, Tome 7 (2001) no. 4, pp. 447-472 | Article | MR 1854032 | Zbl 01901440

[25] Sidorov, N. Arithmetic dynamics, Topics in dynamics and control theory, London Mathematical Society Lecture Note Series, Tome 310 (2003), pp. 145-189 | MR 2052279 | Zbl 1051.37007

[26] Sirvent, V. F.; Wang, Y. Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math., Tome 73 (2002) no. 2, pp. 465-485 | Article | MR 1926787 | Zbl 1048.37015

[27] Solomyak, B. Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, Tome 17 (1997) no. 3, pp. 695-738 | Article | MR 1452190 | Zbl 0884.58062

[28] Thurston, W. P. Groups, tilings and finite state automata , Lectures notes distributed in conjunction with the Colloquium Series, in AMS Colloquium lectures (1989)

[29] Thuswaldner, J. M. Unimodular Pisot Substitutions and Their Associated Tiles (2005) (to appear in J. Théor. Nombres Bordeaux) | Numdam | Zbl 05135401

[30] Veech, W. A. The metric theory of interval exchange transformations I. Generic spectral properties., American Journal of Mathematics, Tome 106 (1984) no. 6, pp. 1331-1359 | Article | MR 765582 | Zbl 0631.28006

[31] Williams, R. F. Classification of one-dimensional attractors, Proc. Symp. Pure Math, Tome 14 (1970), pp. 341-361 | MR 266227 | Zbl 0213.50401