Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to β-shifts
[Représentation géométrique et coïncidence pour pavages associés à une substitution de type Pisot non-unimodulaire réductible avec une application aux beta-shifts]
Baker, Veronica1 ; Barge, Marcy1 ; Kwapisz, Jaroslaw1
1 Montana State University Department of Mathematical Sciences Bozeman MT 59717-2400 (USA)
Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2213-2248.

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.

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.

DOI : 10.5802/aif.2238
Classification : 37B50, 11R06, 28D05
Keywords: Substitution, tilings, pure discrete spectrum spectrum, Pisot
Mot clés : substitution, pavages, spectre purement discret, Pisot
Baker, Veronica 1 ; Barge, Marcy 1 ; Kwapisz, Jaroslaw 1

1 Montana State University Department of Mathematical Sciences Bozeman MT 59717-2400 (USA) 
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     title = {Geometric realization and coincidence for reducible non-unimodular {Pisot} tiling spaces with an application to $\beta $-shifts},
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Baker, Veronica; Barge, Marcy; Kwapisz, Jaroslaw. Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts. Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2213-2248. doi : 10.5802/aif.2238. http://archive.numdam.org/articles/10.5802/aif.2238/

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

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

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

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

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

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

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

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

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

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

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

[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, Volume 12 (1992) no. 4, pp. 713-723 | DOI | MR | Zbl

[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, Volume 6 (1986) no. 4, pp. 529-540 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

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

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