Skip to main content Accessibility help

A combinatorial approach to products of Pisot substitutions



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.



Hide All
[ABB+] 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.
[ABBS08] 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), 377419.
[AD13] Avila, A. and Delecroix, V.. Pisot property for the Brun and fully subtractive algorithms. Preprint, 2013.
[Ada04a] Adamczewski, B.. Répartition des suites (n𝛼) n∈ℕ et substitutions. Acta Arith. 112(1) (2004), 122.
[Ada04b] Adamczewski, B.. Symbolic discrepancy and self-similar dynamics. Ann. Inst. Fourier (Grenoble) 54(7) (2004), 22012234.
[Adl98] Adler, R. L.. Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. (N.S.) 35(1) (1998), 156.
[AFSS10] Adamczewski, B., Frougny, C., Siegel, A. and Steiner, W.. Rational numbers with purely periodic 𝛽-expansion. Bull. Lond. Math. Soc. 42(3) (2010), 538552.
[AG05] Akiyama, S. and Gjini, N.. Connectedness of number theoretic tilings. Discrete Math. Theor. Comput. Sci. 7(1) (2005), 269312 (electronic).
[AI01] Arnoux, P. and Ito, S.. Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8(2) (2001), 181207.
[Aki99] 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. 717.
[Aki00] Akiyama, S.. Cubic Pisot units with finite beta expansions. Algebraic Number Theory and Diophantine Analysis (Graz, 1998). de Gruyter, Berlin, 2000, pp. 1126.
[Aki02] Akiyama, S.. On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Japan 54(2) (2002), 283308.
[AL11] Akiyama, S. and Lee, J.-Y.. Algorithm for determining pure pointedness of self-affine tilings. Adv. Math. 226(4) (2011), 28552883.
[AL14] Akiyama, S. and Lee, J.-Y.. Overlap coincidence to strong coincidence in substitution tiling dynamics. European J. Combin. 39 (2014), 233243.
[AR91] Arnoux, P. and Rauzy, G.. Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119(2) (1991), 199215.
[AW70] 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.
[Bar14] Barge, M.. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Preprint, 2014.
[BBJS13] 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. 5870.
[BBK06] 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), 22132248.
[BD02] Barge, M. and Diamond, B.. Coincidence for substitutions of Pisot type. Bull. Soc. Math. France 130(4) (2002), 619626.
[BD14] 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.
[Ber11] Berthé, V.. Multidimensional Euclidean algorithms, numeration and substitutions. Integers 11B (2011), Paper no. A2, 34.
[BFZ05] 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. 333364.
[BJJP13] 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. 107118.
[BJS12] Berthé, V., Jolivet, T. and Siegel, A.. Substitutive Arnoux–Rauzy sequences have pure discrete spectrum. Unif. Distrib. Theory 7(1) (2012), 173197.
[BJS14] Berthé, V., Jolivet, T. and Siegel, A.. Connectedness of Rauzy fractals associated with Arnoux–Rauzy substitutions. RAIRO Theor. Inform. Appl. 40(3) (2014), 249266.
[BK06] Barge, M. and Kwapisz, J.. Geometric theory of unimodular Pisot substitutions. Amer. J. Math. 128(5) (2006), 12191282.
[Bow78] Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71(1) (1978), 130132.
[Bow08] 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).
[Bre81] Brentjes, A. J.. Multidimensional Continued Fraction Algorithms (Mathematical Centre Tracts, 145) . Mathematisch Centrum, Amsterdam, 1981.
[Bru58] 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. 4564.
[BS05] Berthé, V. and Siegel, A.. Tilings associated with beta-numeration and substitutions. Integers 5(3) (2005), A2, 46 p.
[BST10] 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.
[BST14] Berthé, V., Steiner, W. and Thuswaldner, J.. Tilings with S-adic Rauzy fractals. Preprint, 2014.
[BŠW13] Barge, M., Štimac, S. and Williams, R. F.. Pure discrete spectrum in substitution tiling spaces. Discrete Contin. Dyn. Syst. 33(2) (2013), 579597.
[CS01] Canterini, V. and Siegel, A.. Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353(12) (2001), 51215144.
[CS03] Clark, A. and Sadun, L.. When size matters: subshifts and their related tiling spaces. Ergod. Th. & Dynam. Sys. 23(4) (2003), 10431057.
[DFPLR04] Dubois, E., Farhane, A. and Paysant-Le Roux, R.. The Jacobi–Perron algorithm and Pisot numbers. Acta Arith. 111(3) (2004), 269275.
[Fer06] Fernique, T.. Multidimensional Sturmian sequences and generalized substitutions. Internat. J. Found. Comput. Sci. 17(3) (2006), 575599.
[Fer09] Fernique, T.. Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 42(10) (2009), 22292238.
[FIY13] 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.
[Fog02] Pytheas Fogg, N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794) . Springer, Berlin, 2002.
[FS92] Frougny, C. and Solomyak, B.. Finite beta-expansions. Ergod. Th. & Dynam. Sys. 12(4) (1992), 713723.
[FT06] Fuchs, C. and Tijdeman, R.. Substitutions, abstract number systems and the space filling property. Ann. Inst. Fourier (Grenoble) 56(7) (2006), 23452389.
[HM06] Hubert, P. and Messaoudi, A.. Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals. Acta Arith. 124(1) (2006), 115.
[Hos86] Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6(4) (1986), 529540.
[HS03] Hollander, M. and Solomyak, B.. Two-symbol Pisot substitutions have pure discrete spectrum. Ergod. Th. & Dynam. Sys. 23(2) (2003), 533540.
[IFHY03] 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), 255283.
[IO93] Ito, S. and Ohtsuki, M.. Modified Jacobi–Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16(2) (1993), 441472.
[IO94] Ito, S. and Ohtsuki, M.. Parallelogram tilings and Jacobi–Perron algorithm. Tokyo J. Math. 17(1) (1994), 3358.
[IR06] Ito, S. and Rao, H.. Atomic surfaces, tilings and coincidence. I. Irreducible case. Israel J. Math. 153 (2006), 129155.
[IY07] 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. 171184.
[Jac68] Jacobi, C. G. J.. Allgemeine Theorie der kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird. J. Reine Angew. Math. 69 (1868), 2964.
[KV98] Kenyon, R. and Vershik, A.. Arithmetic construction of sofic partitions of hyperbolic toral automorphisms. Ergod. Th. & Dynam. Sys. 18(2) (1998), 357372.
[Man02] Manning, A.. A Markov partition that reflects the geometry of a hyperbolic toral automorphism. Trans. Amer. Math. Soc. 354(7) (2002), 28492863.
[PD84] Paysant-Le Roux, R. and Dubois, E.. Une application des nombres de Pisot à l’algorithme de Jacobi–Perron. Monatsh. Math. 98(2) (1984), 145155.
[Per07] Perron, O.. Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64(1) (1907), 176.
[Pra99] Praggastis, B.. Numeration systems and Markov partitions from self-similar tilings. Trans. Amer. Math. Soc. 351(8) (1999), 33153349.
[Que10] Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294) , 2nd edn. Springer, Berlin, 2010.
[Rau82] Rauzy, G.. Nombres algébriques et substitutions. Bull. Soc. Math. France 110(2) (1982), 147178.
[Rev91] Reveillès, J.-P.. Géométrie discrète, calculs en nombres entiers et algorithmes. PhD Thesis, Université Louis Pasteur, Strasbourg, 1991.
[Rob04] 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. 81119.
[Sag] The Sage Development Team, Sage mathematics software,
[SAI01] Sano, Y., Arnoux, P. and Ito, S.. Higher dimensional extensions of substitutions and their dual maps. J. Anal. Math. 83 (2001), 183206.
[Sch73] Schweiger, F.. The Metrical Theory of Jacobi–Perron Algorithm (Lecture Notes in Mathematics, 334) . Springer, Berlin, 1973.
[Sch95] Schweiger, F.. Ergodic Theory of Fibred Systems and Metric Number Theory (Oxford Science Publications) . The Clarendon Press–Oxford University Press, New York, 1995.
[Sch00] Schweiger, F.. Multidimensional Continued Fractions (Oxford Science Publications) . Oxford University Press, Oxford, 2000.
[Sid03] Sidorov, N.. Arithmetic dynamics. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Note Series, 310) . Cambridge University Press, Cambridge, 2003, pp. 145189.
[Sie00] Siegel, A.. Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot. PhD Thesis, Université de la Méditerranée, 2000.
[Sol97] Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17(3) (1997), 695738; see also Corrections to: Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 19(6) (1999), 1685.
[ST09] Siegel, A. and Thuswaldner, J. M.. Topological properties of Rauzy fractals. Mém. Soc. Math. Fr. (N.S.) 118 (2009), 140.

Related content

Powered by UNSILO

A combinatorial approach to products of Pisot substitutions



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.