Skip to main content Accessibility help

The Pisot conjecture for $\unicode[STIX]{x1D6FD}$ -substitutions

  • MARCY BARGE (a1)


We prove the Pisot conjecture for $\unicode[STIX]{x1D6FD}$ -substitutions: if $\unicode[STIX]{x1D6FD}$ is a Pisot number, then the tiling dynamical system $(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$ associated with the $\unicode[STIX]{x1D6FD}$ -substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.



Hide All
[A1] Akiyama, S.. Cubic Pisot units with finite beta expansions. Algebraic Number Theory and Diophantine Analysis. Eds. Halter-Koch, F. and Tichy, R. F.. de Gruyter, Berlin, New York, 2000, pp. 1126.
[A2] Akiyama, S.. On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Japan 54(2) (2002), 283308.
[ABBLS] Akiyama, S., Barge, M., Berthé, V., Lee, J.-Y. and Siegel, A.. On the Pisot substitution conjecture. Mathematics of Aperiodic Order (Progress in Mathematics, 309) . Eds. Kellendonk, J., Lenz, D. and Savinien, J.. Birkhauser, Basel, 2015, pp. 3372.
[ARS] Akiyama, S., Rao, H. and Steiner, W.. A certain finiteness property of Pisot number systems. J. Number Theory 107(1) (2004), 135160.
[AP] Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C -algebras. Ergod. Th. & Dynam. Sys. 18 (1998), 509537.
[AS] Akiyama, S. and Sadahiro, T.. A self-similar tiling generated by the minimal Pisot number. Acta Math. Info. Univ. Ostrav. 6 (1998), 926.
[Aus] Auslander, J.. Minimal Flows and Their Extensions (North-Holland Mathematical Studies, 153) . North-Holland, Amsterdam, 1988.
[BBK] 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.
[BL] Baake, M. and Lenz, D.. Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Th. & Dynam. Sys. 24 (2004), 18671893.
[B1] Barge, M.. Pure discrete spectrum for a class of one-dimensional tiling systems. Discrete Contin. Dyn. Syst. Ser. A 36(3) (2016), 11591173.
[B2] Barge, M.. Geometrical and spectral properties of Pisot substitutions. Topology Appl. 205(1) (2016), 2846.
[B3] Barge, M.. Factors of Pisot tiling spaces and the coincidence rank conjecture. Bull. Soc. Math. France 143(2) (2015), 357381.
[BD1] Barge, M. and Diamond, B.. A complete invariant for the topology of one-dimensional substitution tiling spaces. Ergod. Th. & Dynam. Sys. 21 (2001), 13331358.
[BD2] Barge, M. and Diamond, B.. Coincidence for substitutions of Pisot type. Bull. Soc. Math. France 130 (2002), 619–626.
[BG] Barge, M. and Gambaudo, J.-M.. Geometric realization for substitution tilings. Ergod. Th. & Dynam. Sys. 34(2) (2014), 457482.
[BK] Barge, M. and Kellendonk, J.. Proximality and pure point spectrum for tiling dynamical systems. Michigan Math. J. 62(4) (2013), 793822.
[BKw] Barge, M. and Kwapisz, J.. Geometric theory of unimodular Pisot substitutions. Amer. J. Math. 128 (2006), 12191282.
[BO] Barge, M. and Olimb, C.. Asymptotic structure in substitution tiling spaces. Ergod. Th. & Dynam. Sys. 34(1) (2014), 5594.
[BSW] Barge, M., Štimac, S. and Williams, R. F.. Pure discrete spectrum in substitution tiling spaces. Discrete. Contin. Dyn. Syst. Ser. A 2 (2013), 579597.
[BS] Berthé, V. and Siegel, A.. Tilings associated with beta-numeration and substitutions. Integers 5 (2005), A02.
[B-M] Bertrand-Mathis, A.. Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285(6) (1977), A419A421.
[Bl] Blanchard, F.. 𝛽-expansion and symbolic dynamics. Theoret. Comput. Sci. 65 (1989), 131141.
[Bor] le Borgne, S.. Un codage sofique des automorphismes hyperboliques du tore. C. R. Acad. Sci. Paris 323 (1996), 11231128.
[CS] Clark, A. and Sadun, L.. When size matters: subshifts and their related tiling spaces. Ergod. Th. & Dynam. Sys. 23 (2003), 10431057.
[EI] Ei, H. and Ito, S.. Tilings from some non-irreducible, Pisot substitutions. Discrete Math. Theor. Comput. Sci. 8(1) (2005), 81122.
[D] Dworkin, S.. Spectral theory and x-ray diffraction. J. Math. Phys. 34 (1993), 29642967.
[Dur] Durand, F.. A characterization of substitutive sequences using return words. Discrete Math. 179 (1998), 89101.
[FS] Frougny, C. and Solomyak, B.. Finite beta-expansions. Ergod. Th. & Dynam. Sys. 12 (1992), 713723.
[Hof1] Hofbauer, F.. 𝛽-shifts have unique maximal measure. Monatsh. Math. 85 (1978), 189198.
[Hof2] Hofbauer, F.. Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 52 (1980), 289300.
[H] Hollander, M.. Linear Numeration systems, finite beta expansions, and discrete spectrum of substitution dynamical systems. PhD Thesis, University of Washington, 1996.
[IT] Ito, S. and Takahashi, Y.. Markov subshifts and realization of 𝛽-expansions. J. Math Soc. Japan 26(1) (1974), 3355.
[KS] Kalle, C. and Steiner, W.. Beta-expansions, natural extensions and multiple tilings associated with Pisot units. Trans. Amer. Math. Soc. 364(5) (2012), 22812318.
[KV] Kenyon, R. and Vershik, A.. Arithmetic construction of sofic partitions and hyperbolic toral automorphisms. Ergod. Th. & Dynam. Sys. 18 (1998), 357372.
[Le] Lenz, D.. Aperiodic order and pure point diffraction. Philos. Mag. 88(13–15) (2008), 20592071.
[LM] D., Lind and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.
[LMS] Lee, J.-Y., Moody, R. and Solomyak, B.. Consequences of pure point diffraction spectra for multiset substitution systems. Discrete Comput. Geom. 29 (2003), 525560.
[LS] Lee, J.-Y. and Solomyak, B.. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete Contin. Dyn. Syst. Ser. A 32(3) (2012), 935959.
[Pa] Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Hungar. 11 (1960), 269278.
[Pr] Praggastis, B.. Markov partition for hyperbolic toral automorphism. PhD Thesis, University of Washington, 1992.
[Q] Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294) . Springer, Berlin, 1987.
[Ra] Rauzy, G.. Nombres Algébriques et substitutions. Bull. Soc. Math. France 110 (1982), 147178.
[R] Rényi, A.. Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungary 8 (1957), 477493.
[Ro] Robinson, E. A. Jr. Symbolic dynamics and tilings of Rd. Symbolic Dynamics and its Applications (Proceedings of Symposia in Applied Mathematics, 60) . American Mathematical Society, Providence, RI, 2004, pp. 81119.
[Sch1] Schmidt, K.. On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12 (1980), 269278.
[Sch2] Schmidt, K.. Algebraic codings of expansive group automorphisms and two-sided beta-shifts. Monatsh. Math. 129 (2000), 3761.
[Si1] Sidorov, N.. Bijective and general arithmetic codings for Pisot toral automorphisms. J. Dyn. Control Syst. 7(4) (2001), 447472.
[Si2] Sidorov, N.. Arithmetic dynamics. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Note Series, 310) . Cambridge University Press, Cambridge, 2003, pp. 145189.
[S1] Solomyak, B.. Substitutions, adic transformations and beta-expansions. Contemp. Math. 135 (1992), 361372.
[S2] Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695738.
[S3] Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), 265279.
[S4] Solomyak, B.. Eigenfunctions for substitution tiling systems. Adv. Stud. Pure Math. 49 (2007), 433454.
[T] Thurston, W. P.. Groups, Tilings and Finite State Automata (AMS Colloquium Lectures) . American Mathematical Society, Boulder, CO, 1989.
[V] Veech, W. A.. The equicontinuous structure relation for minimal Abelian transformation groups. Amer. J. Math. 90 (1968), 723732.
[Ver1] Vershik, A.. The fibadic expansion of real numbers and adic transformations. Preprint, Mittag-Leffler Institute, 1991/92.
[Ver2] Vershik, A.. Arithmetic isomorphism of the toral hyperbolic automorphisms and sofic systems. Funct. Anal. Appl. 26 (1992), 170173.

Related content

Powered by UNSILO

The Pisot conjecture for $\unicode[STIX]{x1D6FD}$ -substitutions

  • MARCY BARGE (a1)


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.