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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

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 & 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.  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), 509-537. doi: 10.1017/S0143385798100457.  Google Scholar

[3]

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

[4]

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

[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.  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), 2213-2248. 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), 1219-1282. 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), A02.arXiv:1108.5574.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 577-593.  Google Scholar

[21]

J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets, Journal of Geometry and Physics, 57 (2007), 2263-2285. 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), 173-201. 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), 525-560. 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), 319-338. 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), 83-104.  Google Scholar

[27]

P. Michel, Coincidence values and spectra of substitutions, Zeit. Wahr., 42 (1978), 205-227. 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), 697-710. Dedicated to Robert V. Moody. 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), 265-279. doi: 10.1007/PL00009386.  Google Scholar

[31]

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

[32]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory & Dynamical Systems, 17 (1997), 695-738. 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), 723-732. 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), 2855-2883. 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), 509-537. doi: 10.1017/S0143385798100457.  Google Scholar

[3]

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

[4]

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

[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.  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), 2213-2248. 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), 1219-1282. 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), A02.arXiv:1108.5574.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 577-593.  Google Scholar

[21]

J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets, Journal of Geometry and Physics, 57 (2007), 2263-2285. 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), 173-201. 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), 525-560. 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), 319-338. 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), 83-104.  Google Scholar

[27]

P. Michel, Coincidence values and spectra of substitutions, Zeit. Wahr., 42 (1978), 205-227. 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), 697-710. Dedicated to Robert V. Moody. 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), 265-279. doi: 10.1007/PL00009386.  Google Scholar

[31]

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

[32]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory & Dynamical Systems, 17 (1997), 695-738. 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), 723-732. Google Scholar

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