# American Institute of Mathematical Sciences

February  2013, 33(2): 579-597. doi: 10.3934/dcds.2013.33.579

## Pure discrete spectrum in substitution tiling spaces

 1 Department of Mathematics, Montana State University, Bozeman, MT 59717, United States 2 Department of Mathematics, University of Zagreb, Bijenička 30, 10 000 Zagreb 3 Department of Mathematics, University of Texas, Austin, TX 78712, United States

Received  July 2011 Revised  April 2012 Published  September 2012

We introduce a technique for establishing pure discrete spectrum for substitution tiling systems of Pisot family type and illustrate with several examples.
Citation: Marcy Barge, Sonja Štimac, R. F. Williams. Pure discrete spectrum in substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 579-597. doi: 10.3934/dcds.2013.33.579
##### References:
 [1] S. Akiyama and J. Y. Lee, Algorithm for determining pure pointedness of self- affine tilings,, Adv. Math., 226 (2011), 2855.  doi: 10.1016/j.aim.2010.07.019.  Google Scholar [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.  doi: 10.1017/S0143385798100457.  Google Scholar [3] P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals,, Bull. Belg. Math Soc., 8 (2001), 18.   Google Scholar [4] J. Auslander, "Minimal Flows and Their Extensions,", North-Holland Mathematical Studies, (1988).   Google Scholar [5] M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction,, J. Reine Angew. Math., 573 (2004), 61.  doi: 10.1515/crll.2004.064.  Google Scholar [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.  doi: 10.5802/aif.2238.  Google Scholar [7] M. Barge, H. Bruin, L. Jones and L. Sadun, Homological Pisot substitutions and exact regularity,, To appear in Israel J. Math., ().   Google Scholar [8] M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems,, preprint, ().   Google Scholar [9] M. Barge, J. Kellendonk and S. Schmeiding, Maximal equicontinuous factors and cohomology of tiling spaces,, To appear in Fund. Math., ().   Google Scholar [10] M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions,, Amer J. Math., 128 (2006), 1219.  doi: 10.1353/ajm.2006.0037.  Google Scholar [11] M. Barge and C. Olimb, Asymptotic structure in substitution tiling spaces,, To appear in Ergodic Theory & Dynamical Systems, ().   Google Scholar [12] V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum,, preprint, ().   Google Scholar [13] V. Berthé and A. Siegel, Tilings associated with beta-numeration and substitutions,, Integers: Electronic Journal of Combinatorial Number Theory, 5 (2005).   Google Scholar [14] F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length,, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41 (1978), 221.   Google Scholar [15] S. Dworkin, Spectral theory and X-ray diffraction,, J. Math. Phys., 34 (1993), 2965.  doi: 10.1063/1.530108.  Google Scholar [16] N. P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", Lecture notes in mathematics, (2002).   Google Scholar [17] D. Fretlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets,, Discrete Comput. Geom., 37 (2007), 381.  doi: 10.1007/s00454-006-1280-9.  Google Scholar [18] S. Ito and H. Rao, Atomic surfaces, tiling and coincidence I. Irreducible case,, Israel J. Math., 153 (2006), 129.  doi: 10.1007/BF02771781.  Google Scholar [19] R. Kenyon, Ph. D. Thesis,, Princeton University, (1990).   Google Scholar [20] R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings,, Discrete Comput. Geom., 43 (2010), 577.   Google Scholar [21] J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets,, Journal of Geometry and Physics, 57 (2007), 2263.  doi: 10.1016/j.geomphys.2007.07.003.  Google Scholar [22] J. Y. Lee and R. Moody, Lattice substitution systems and model sets,, Discrete Comput. Geom., 25 (2001), 173.  doi: 10.1007/s004540010083.  Google Scholar [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.  doi: 10.1007/s00454-003-0781-z.  Google Scholar [24] J. Y. Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comp. Geom., 34 (2008), 319.  doi: 10.1007/s00454-008-9054-1.  Google Scholar [25] J. Y. Lee and B. Solomyak, Pisot family self-affine tilings, discrete spectrum, and the Meyer property,, preprint, ().   Google Scholar [26] A. N. Livshits, Some examples of adic transformations and substitutions,, Selecta Math. Sovietica, 11 (1992), 83.   Google Scholar [27] P. Michel, Coincidence values and spectra of substitutions,, Zeit. Wahr., 42 (1978), 205.  doi: 10.1007/BF00641410.  Google Scholar [28] A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals,, preprint., ().   Google Scholar [29] V. F. Sirivent and B. Solomyak, Pure discrete spectrum for one-dimensional substitution systems of Pisot type,, Canad. Math. Bull., 45 (2002), 697.  doi: 10.4153/CMB-2002-062-3.  Google Scholar [30] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geometry, 20 (1998), 265.  doi: 10.1007/PL00009386.  Google Scholar [31] B. Solomyak, Eigenfunctions for substitution tiling systems,, Advanced Studies in Pure Mathematics, 49 (2007), 433.   Google Scholar [32] B. Solomyak, Dynamics of self-similar tilings,, Ergodic Theory & Dynamical Systems, 17 (1997), 695.  doi: 10.1017/S0143385797084988.  Google Scholar [33] W. A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups,, Amer. J. of Math. 90 (1968), 90 (1968), 723.   Google Scholar

show all references

##### References:
 [1] S. Akiyama and J. Y. Lee, Algorithm for determining pure pointedness of self- affine tilings,, Adv. Math., 226 (2011), 2855.  doi: 10.1016/j.aim.2010.07.019.  Google Scholar [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.  doi: 10.1017/S0143385798100457.  Google Scholar [3] P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals,, Bull. Belg. Math Soc., 8 (2001), 18.   Google Scholar [4] J. Auslander, "Minimal Flows and Their Extensions,", North-Holland Mathematical Studies, (1988).   Google Scholar [5] M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction,, J. Reine Angew. Math., 573 (2004), 61.  doi: 10.1515/crll.2004.064.  Google Scholar [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.  doi: 10.5802/aif.2238.  Google Scholar [7] M. Barge, H. Bruin, L. Jones and L. Sadun, Homological Pisot substitutions and exact regularity,, To appear in Israel J. Math., ().   Google Scholar [8] M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems,, preprint, ().   Google Scholar [9] M. Barge, J. Kellendonk and S. Schmeiding, Maximal equicontinuous factors and cohomology of tiling spaces,, To appear in Fund. Math., ().   Google Scholar [10] M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions,, Amer J. Math., 128 (2006), 1219.  doi: 10.1353/ajm.2006.0037.  Google Scholar [11] M. Barge and C. Olimb, Asymptotic structure in substitution tiling spaces,, To appear in Ergodic Theory & Dynamical Systems, ().   Google Scholar [12] V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum,, preprint, ().   Google Scholar [13] V. Berthé and A. Siegel, Tilings associated with beta-numeration and substitutions,, Integers: Electronic Journal of Combinatorial Number Theory, 5 (2005).   Google Scholar [14] F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length,, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41 (1978), 221.   Google Scholar [15] S. Dworkin, Spectral theory and X-ray diffraction,, J. Math. Phys., 34 (1993), 2965.  doi: 10.1063/1.530108.  Google Scholar [16] N. P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", Lecture notes in mathematics, (2002).   Google Scholar [17] D. Fretlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets,, Discrete Comput. Geom., 37 (2007), 381.  doi: 10.1007/s00454-006-1280-9.  Google Scholar [18] S. Ito and H. Rao, Atomic surfaces, tiling and coincidence I. Irreducible case,, Israel J. Math., 153 (2006), 129.  doi: 10.1007/BF02771781.  Google Scholar [19] R. Kenyon, Ph. D. Thesis,, Princeton University, (1990).   Google Scholar [20] R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings,, Discrete Comput. Geom., 43 (2010), 577.   Google Scholar [21] J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets,, Journal of Geometry and Physics, 57 (2007), 2263.  doi: 10.1016/j.geomphys.2007.07.003.  Google Scholar [22] J. Y. Lee and R. Moody, Lattice substitution systems and model sets,, Discrete Comput. Geom., 25 (2001), 173.  doi: 10.1007/s004540010083.  Google Scholar [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.  doi: 10.1007/s00454-003-0781-z.  Google Scholar [24] J. Y. Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comp. Geom., 34 (2008), 319.  doi: 10.1007/s00454-008-9054-1.  Google Scholar [25] J. Y. Lee and B. Solomyak, Pisot family self-affine tilings, discrete spectrum, and the Meyer property,, preprint, ().   Google Scholar [26] A. N. Livshits, Some examples of adic transformations and substitutions,, Selecta Math. Sovietica, 11 (1992), 83.   Google Scholar [27] P. Michel, Coincidence values and spectra of substitutions,, Zeit. Wahr., 42 (1978), 205.  doi: 10.1007/BF00641410.  Google Scholar [28] A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals,, preprint., ().   Google Scholar [29] V. F. Sirivent and B. Solomyak, Pure discrete spectrum for one-dimensional substitution systems of Pisot type,, Canad. Math. Bull., 45 (2002), 697.  doi: 10.4153/CMB-2002-062-3.  Google Scholar [30] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geometry, 20 (1998), 265.  doi: 10.1007/PL00009386.  Google Scholar [31] B. Solomyak, Eigenfunctions for substitution tiling systems,, Advanced Studies in Pure Mathematics, 49 (2007), 433.   Google Scholar [32] B. Solomyak, Dynamics of self-similar tilings,, Ergodic Theory & Dynamical Systems, 17 (1997), 695.  doi: 10.1017/S0143385797084988.  Google Scholar [33] W. A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups,, Amer. J. of Math. 90 (1968), 90 (1968), 723.   Google Scholar
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