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Pure discrete spectrum in substitution tiling spaces

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  • We introduce a technique for establishing pure discrete spectrum for substitution tiling systems of Pisot family type and illustrate with several examples.
    Mathematics Subject Classification: Primary: 37B50, 52C22, 52C23; Secondary: 37B05, 11R06.

    Citation:

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  • [1]

    S. Akiyama and J. Y. Lee, Algorithm for determining pure pointedness of self- affine tilings, Adv. Math., 226 (2011), 2855-2883.doi: 10.1016/j.aim.2010.07.019.

    [2]

    J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $c^*$-algebras, Ergodic Theory & Dynamical Systems, 18 (1998), 509-537.doi: 10.1017/S0143385798100457.

    [3]

    P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 18-2007.

    [4]

    J. Auslander, "Minimal Flows and Their Extensions," North-Holland Mathematical Studies, 153, North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1988.

    [5]

    M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math., 573 (2004), 61-94. .doi: 10.1515/crll.2004.064.

    [6]

    V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta$-shifts, J. Instit. Fourier, 56 (2006), 2213-2248.doi: 10.5802/aif.2238.

    [7]

    M. Barge, H. Bruin, L. Jones and L. SadunHomological Pisot substitutions and exact regularity, To appear in Israel J. Math., preprint, arXiv:1001.2027.

    [8]

    M. Barge and J. KellendonkProximality and pure point spectrum for tiling dynamical systems, preprint, arXiv:1108.4065.

    [9]

    M. Barge, J. Kellendonk and S. SchmeidingMaximal equicontinuous factors and cohomology of tiling spaces, To appear in Fund. Math.,arXiv:1204.1432.

    [10]

    M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., 128 (2006), 1219-1282.doi: 10.1353/ajm.2006.0037.

    [11]

    M. Barge and C. OlimbAsymptotic structure in substitution tiling spaces, To appear in Ergodic Theory & Dynamical Systems, preprint, arXiv:1101.4902.

    [12]

    V. Berthé, T. Jolivet and A. SiegelSubstitutive Arnoux-Rauzy substitutions have pure discrete spectrum, preprint, arXiv:1108.5574.

    [13]

    V. Berthé and A. Siegel, Tilings associated with beta-numeration and substitutions, Integers: Electronic Journal of Combinatorial Number Theory, 5 (2005), A02.arXiv:1108.5574.

    [14]

    F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41 (1978), 221-239.

    [15]

    S. Dworkin, Spectral theory and X-ray diffraction, J. Math. Phys., 34 (1993), 2965-2967.doi: 10.1063/1.530108.

    [16]

    N. P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," Lecture notes in mathematics, (eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel), Springer-Verlag, 2002.

    [17]

    D. Fretlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets, Discrete Comput. Geom., 37 (2007), 381-407.doi: 10.1007/s00454-006-1280-9.

    [18]

    S. Ito and H. Rao, Atomic surfaces, tiling and coincidence I. Irreducible case, Israel J. Math., 153 (2006), 129-156.doi: 10.1007/BF02771781.

    [19]

    R. Kenyon, Ph. D. Thesis, Princeton University, 1990.

    [20]

    R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings, Discrete Comput. Geom., 43 (2010), 577-593.

    [21]

    J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets, Journal of Geometry and Physics, 57 (2007), 2263-2285.doi: 10.1016/j.geomphys.2007.07.003.

    [22]

    J. Y. Lee and R. Moody, Lattice substitution systems and model sets, Discrete Comput. Geom., 25 (2001), 173-201.doi: 10.1007/s004540010083.

    [23]

    J. Y. Lee, R. Moody and B. Solomyak, Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems, Discrete Comp. Geom., 29 (2003), 525-560.doi: 10.1007/s00454-003-0781-z.

    [24]

    J. Y. Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comp. Geom., 34 (2008), 319-338.doi: 10.1007/s00454-008-9054-1.

    [25]

    J. Y. Lee and B. SolomyakPisot family self-affine tilings, discrete spectrum, and the Meyer property, preprint, arXiv:1002.0039.

    [26]

    A. N. Livshits, Some examples of adic transformations and substitutions, Selecta Math. Sovietica, 11 (1992), 83-104.

    [27]

    P. Michel, Coincidence values and spectra of substitutions, Zeit. Wahr., 42 (1978), 205-227.doi: 10.1007/BF00641410.

    [28]

    A. Siegel and J. ThuswaldnerTopological properties of Rauzy fractals, preprint.

    [29]

    V. F. Sirivent and B. Solomyak, Pure discrete spectrum for one-dimensional substitution systems of Pisot type, Canad. Math. Bull., 45 (2002), 697-710. Dedicated to Robert V. Moody.doi: 10.4153/CMB-2002-062-3.

    [30]

    B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279.doi: 10.1007/PL00009386.

    [31]

    B. Solomyak, Eigenfunctions for substitution tiling systems, Advanced Studies in Pure Mathematics, 49 (2007), 433-454.

    [32]

    B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory & Dynamical Systems, 17 (1997), 695-738.doi: 10.1017/S0143385797084988.

    [33]

    W. A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. of Math. 90 (1968), 723-732.

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