Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T15:29:45.892Z Has data issue: false hasContentIssue false
  • English
  • Français

A combinatorial approach to products of Pisot substitutions

Published online by Cambridge University Press:  19 March 2015

VALÉRIE BERTHÉ ,
JÉRÉMIE BOURDON ,
TIMO JOLIVET  and
ANNE SIEGEL
VALÉRIE BERTHÉ
Affiliation:
LIAFA, CNRS, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France email berthe@liafa.univ-paris-diderot.fr
JÉRÉMIE BOURDON
Affiliation:
LINA, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex, France
TIMO JOLIVET
Affiliation:
LIAFA, CNRS, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France email berthe@liafa.univ-paris-diderot.fr FUNDIM, Department of Mathematics, University of Turku, 20014 Turku, Finland
ANNE SIEGEL
Affiliation:
INRIA, CNRS, Université de Rennes 1, IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 6 , September 2016 , pp. 1757 - 1794
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akiyama, S., Barge, M., Berthé, V., Lee, J.-Y. and Siegel, A.. On the Pisot substitution conjecture. Mathematics of Aperiodic Order (Progress in Mathematics). Eds. J. Kellendonk, D. Lenz and J. Savinien. Birkhäuser, to appear.Google Scholar
Akiyama, S., Barat, G., Berthé, V. and Siegel, A.. Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. Monatsh. Math. 155(3–4) (2008), 377–419.CrossRefGoogle Scholar
Avila, A. and Delecroix, V.. Pisot property for the Brun and fully subtractive algorithms. Preprint, 2013.Google Scholar
Adamczewski, B.. Répartition des suites (n𝛼) n∈ℕ et substitutions. Acta Arith. 112(1) (2004), 1–22.Google Scholar
Adamczewski, B.. Symbolic discrepancy and self-similar dynamics. Ann. Inst. Fourier (Grenoble) 54(7) (2004), 2201–2234.Google Scholar
Adler, R. L.. Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. (N.S.) 35(1) (1998), 1–56.Google Scholar
Adamczewski, B., Frougny, C., Siegel, A. and Steiner, W.. Rational numbers with purely periodic 𝛽-expansion. Bull. Lond. Math. Soc. 42(3) (2010), 538–552.Google Scholar
Akiyama, S. and Gjini, N.. Connectedness of number theoretic tilings. Discrete Math. Theor. Comput. Sci. 7(1) (2005), 269–312 (electronic).Google Scholar
Arnoux, P. and Ito, S.. Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8(2) (2001), 181–207.Google Scholar
Akiyama, S.. Self affine tiling and Pisot numeration system. Number Theory and its Applications (Kyoto, 1997) (Developments in Mathematics, 2) . Kluwer Academic, Dordrecht, 1999, pp. 7–17.Google Scholar
Akiyama, S.. Cubic Pisot units with finite beta expansions. Algebraic Number Theory and Diophantine Analysis (Graz, 1998). de Gruyter, Berlin, 2000, pp. 11–26.Google Scholar
Akiyama, S.. On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Japan 54(2) (2002), 283–308.Google Scholar
Akiyama, S. and Lee, J.-Y.. Algorithm for determining pure pointedness of self-affine tilings. Adv. Math. 226(4) (2011), 2855–2883.Google Scholar
Akiyama, S. and Lee, J.-Y.. Overlap coincidence to strong coincidence in substitution tiling dynamics. European J. Combin. 39 (2014), 233–243.Google Scholar
Arnoux, P. and Rauzy, G.. Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119(2) (1991), 199–215.CrossRefGoogle Scholar
Adler, R. L. and Weiss, B.. Similarity of Automorphisms of the Torus (Memoirs of the American Mathematical Society, 98) . American Mathematical Society, Providence, RI, 1970.Google Scholar
Barge, M.. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Preprint, 2014.Google Scholar
Berthé, V., Bourdon, J., Jolivet, T. and Siegel, A.. Generating discrete planes with substitutions. Combinatorics on Words, 9th Int. Conf. (Lecture Notes in Computer Science, 8079) . Eds. Karhumäki, J., Lepistö, A. and Zamboni, L. Q.. Springer, Heidelberg, 2013, pp. 58–70.Google Scholar
Baker, V., Barge, M. and Kwapisz, J.. Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to 𝛽-shifts. Ann. Inst. Fourier (Grenoble) 56(7) (2006), 2213–2248.Google Scholar
Barge, M. and Diamond, B.. Coincidence for substitutions of Pisot type. Bull. Soc. Math. France 130(4) (2002), 619–626.Google Scholar
Berthé, V. and Delecroix, V.. Beyond substitutive dynamical systems: S-adic expansions. Proc. RIMS Conf. Numeration and Substitution 2012, B46, 2014, pp. 81–123, to appear.Google Scholar
Berthé, V.. Multidimensional Euclidean algorithms, numeration and substitutions. Integers 11B (2011), Paper no. A2, 34.Google Scholar
Berthé, V., Ferenczi, S. and Zamboni, L. Q.. Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly. Algebraic and Topological Dynamics (Contemporary Mathematics, 385) . American Mathematical Society, Providence, RI, 2005, pp. 333–364.Google Scholar
Berthé, V., Jamet, D., Jolivet, T. and Provençal, X.. Critical connectedness of thin arithmetical discrete planes. Discrete Geometry for Computer Imagery, 17th IAPR Int. Conf. (Lecture Notes in Computer Science, 7749) . Eds. Gonzaléz-Díaz, R., Jiménez, M. J. and Medrano, B.. Springer, Heidelberg, 2013, pp. 107–118.Google Scholar
Berthé, V., Jolivet, T. and Siegel, A.. Substitutive Arnoux–Rauzy sequences have pure discrete spectrum. Unif. Distrib. Theory 7(1) (2012), 173–197.Google Scholar
Berthé, V., Jolivet, T. and Siegel, A.. Connectedness of Rauzy fractals associated with Arnoux–Rauzy substitutions. RAIRO Theor. Inform. Appl. 40(3) (2014), 249–266.Google Scholar
Barge, M. and Kwapisz, J.. Geometric theory of unimodular Pisot substitutions. Amer. J. Math. 128(5) (2006), 1219–1282.Google Scholar
Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71(1) (1978), 130–132.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) , revised edn. Springer, Berlin, 2008 (with a preface by D. Ruelle, edited by J.-R. Chazottes).Google Scholar
Brentjes, A. J.. Multidimensional Continued Fraction Algorithms (Mathematical Centre Tracts, 145) . Mathematisch Centrum, Amsterdam, 1981.Google Scholar
Brun, V.. Algorithmes euclidiens pour trois et quatre nombres. Treizième congrès des mathématiciens scandinaves, tenu à Helsinki 18–23 août 1957. Mercators Tryckeri, Helsinki, 1958, pp. 45–64.Google Scholar
Berthé, V. and Siegel, A.. Tilings associated with beta-numeration and substitutions. Integers 5(3) (2005), A2, 46 p.Google Scholar
Berthé, V., Siegel, A. and Thuswaldner, J. M.. Substitutions, Rauzy fractals, and tilings. Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and its Applications, 135) . Cambridge University Press, Cambridge, 2010.Google Scholar
Berthé, V., Steiner, W. and Thuswaldner, J.. Tilings with S-adic Rauzy fractals. Preprint, 2014.Google Scholar
Barge, M., Štimac, S. and Williams, R. F.. Pure discrete spectrum in substitution tiling spaces. Discrete Contin. Dyn. Syst. 33(2) (2013), 579–597.Google Scholar
Canterini, V. and Siegel, A.. Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353(12) (2001), 5121–5144.Google Scholar
Clark, A. and Sadun, L.. When size matters: subshifts and their related tiling spaces. Ergod. Th. & Dynam. Sys. 23(4) (2003), 1043–1057.Google Scholar
Dubois, E., Farhane, A. and Paysant-Le Roux, R.. The Jacobi–Perron algorithm and Pisot numbers. Acta Arith. 111(3) (2004), 269–275.Google Scholar
Fernique, T.. Multidimensional Sturmian sequences and generalized substitutions. Internat. J. Found. Comput. Sci. 17(3) (2006), 575–599.Google Scholar
Fernique, T.. Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 42(10) (2009), 2229–2238.Google Scholar
Furukado, M., Ito, S. and Yasutomi, S.-I.. The condition for the generation of the stepped surfaces in terms of the modified Jacobi–Perron algorithm. Preprint, 2013.Google Scholar
Pytheas Fogg, N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794) . Springer, Berlin, 2002.Google Scholar
Frougny, C. and Solomyak, B.. Finite beta-expansions. Ergod. Th. & Dynam. Sys. 12(4) (1992), 713–723.Google Scholar
Fuchs, C. and Tijdeman, R.. Substitutions, abstract number systems and the space filling property. Ann. Inst. Fourier (Grenoble) 56(7) (2006), 2345–2389.Google Scholar
Hubert, P. and Messaoudi, A.. Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals. Acta Arith. 124(1) (2006), 1–15.Google Scholar
Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6(4) (1986), 529–540.Google Scholar
Hollander, M. and Solomyak, B.. Two-symbol Pisot substitutions have pure discrete spectrum. Ergod. Th. & Dynam. Sys. 23(2) (2003), 533–540.Google Scholar
Ito, S., Fujii, J., Higashino, H. and Yasutomi, S.-I.. On simultaneous approximation to (𝛼, 𝛼2 with 𝛼3 + k𝛼 - 1 = 0. J. Number Theory 99(2) (2003), 255–283.Google Scholar
Ito, S. and Ohtsuki, M.. Modified Jacobi–Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16(2) (1993), 441–472.Google Scholar
Ito, S. and Ohtsuki, M.. Parallelogram tilings and Jacobi–Perron algorithm. Tokyo J. Math. 17(1) (1994), 33–58.Google Scholar
Ito, S. and Rao, H.. Atomic surfaces, tilings and coincidence. I. Irreducible case. Israel J. Math. 153 (2006), 129–155.Google Scholar
Ito, S. and Yasutomi, S.-I.. On simultaneous Diophantine approximation to periodic points related to modified Jacobi–Perron algorithm. Probability and Number Theory—Kanazawa 2005 (Advanced Studies in Pure Mathematics, 49) . Mathematical Society of Japan, Tokyo, 2007, pp. 171–184.CrossRefGoogle Scholar
Jacobi, C. G. J.. Allgemeine Theorie der kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird. J. Reine Angew. Math. 69 (1868), 29–64.Google Scholar
Kenyon, R. and Vershik, A.. Arithmetic construction of sofic partitions of hyperbolic toral automorphisms. Ergod. Th. & Dynam. Sys. 18(2) (1998), 357–372.Google Scholar
Manning, A.. A Markov partition that reflects the geometry of a hyperbolic toral automorphism. Trans. Amer. Math. Soc. 354(7) (2002), 2849–2863.Google Scholar
Paysant-Le Roux, R. and Dubois, E.. Une application des nombres de Pisot à l’algorithme de Jacobi–Perron. Monatsh. Math. 98(2) (1984), 145–155.Google Scholar
Perron, O.. Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64(1) (1907), 1–76.Google Scholar
Praggastis, B.. Numeration systems and Markov partitions from self-similar tilings. Trans. Amer. Math. Soc. 351(8) (1999), 3315–3349.Google Scholar
Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294) , 2nd edn. Springer, Berlin, 2010.Google Scholar
Rauzy, G.. Nombres algébriques et substitutions. Bull. Soc. Math. France 110(2) (1982), 147–178.Google Scholar
Reveillès, J.-P.. Géométrie discrète, calculs en nombres entiers et algorithmes. PhD Thesis, Université Louis Pasteur, Strasbourg, 1991.Google Scholar
Robinson, E. A. Jr. Symbolic dynamics and tilings of ℝ d . Symbolic Dynamics and its Applications (Proceedings of Symposia in Applied Mathematics, 60) . American Mathematical Society, Providence, RI, 2004, pp. 81–119.CrossRefGoogle Scholar
The Sage Development Team, Sage mathematics software, http://www.sagemath.org.Google Scholar
Sano, Y., Arnoux, P. and Ito, S.. Higher dimensional extensions of substitutions and their dual maps. J. Anal. Math. 83 (2001), 183–206.CrossRefGoogle Scholar
Schweiger, F.. The Metrical Theory of Jacobi–Perron Algorithm (Lecture Notes in Mathematics, 334) . Springer, Berlin, 1973.CrossRefGoogle Scholar
Schweiger, F.. Ergodic Theory of Fibred Systems and Metric Number Theory (Oxford Science Publications) . The Clarendon Press–Oxford University Press, New York, 1995.Google Scholar
Schweiger, F.. Multidimensional Continued Fractions (Oxford Science Publications) . Oxford University Press, Oxford, 2000.Google Scholar
Sidorov, N.. Arithmetic dynamics. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Note Series, 310) . Cambridge University Press, Cambridge, 2003, pp. 145–189.Google Scholar
Siegel, A.. Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot. PhD Thesis, Université de la Méditerranée, 2000.Google Scholar
Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17(3) (1997), 695–738; see also Corrections to: Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 19(6) (1999), 1685.CrossRefGoogle Scholar
Siegel, A. and Thuswaldner, J. M.. Topological properties of Rauzy fractals. Mém. Soc. Math. Fr. (N.S.) 118 (2009), 140.Google Scholar