Pure discrete spectrum in substitution tiling spaces

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

Pure discrete spectrum in substitution tiling spaces

Marcy Barge ^1, , Sonja Štimac ^2, and R. F. Williams ^3,

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

Abstract
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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.

Keywords: Pure discrete spectrum, substitution, maximal equicontinuous factor, coincidence., tiling space.

Mathematics Subject Classification: Primary: 37B50, 52C22, 52C23; Secondary: 37B05, 11R0.

Citation: Marcy Barge, Sonja Štimac, R. F. Williams. Pure discrete spectrum in substitution tiling spaces. Discrete and Continuous Dynamical Systems, 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), 2855-2883. 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. Sadun, Homological Pisot substitutions and exact regularity, To appear in Israel J. Math., preprint, arXiv:1001.2027.
[8]	M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, preprint, arXiv:1108.4065.
[9]	M. Barge, J. Kellendonk and S. Schmeiding, Maximal 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. Olimb, Asymptotic structure in substitution tiling spaces, To appear in Ergodic Theory & Dynamical Systems, preprint, arXiv:1101.4902.
[12]	V. Berthé, T. Jolivet and A. Siegel, Substitutive 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. Solomyak, Pisot 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. Thuswaldner, Topological 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.

show all references

References:

[1]	S. Akiyama and J. Y. Lee, Algorithm for determining pure pointedness of self- affine tilings, Adv. Math., 226 (2011), 2855-2883. 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. Sadun, Homological Pisot substitutions and exact regularity, To appear in Israel J. Math., preprint, arXiv:1001.2027.
[8]	M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, preprint, arXiv:1108.4065.
[9]	M. Barge, J. Kellendonk and S. Schmeiding, Maximal 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. Olimb, Asymptotic structure in substitution tiling spaces, To appear in Ergodic Theory & Dynamical Systems, preprint, arXiv:1101.4902.
[12]	V. Berthé, T. Jolivet and A. Siegel, Substitutive 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. Solomyak, Pisot 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. Thuswaldner, Topological 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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