April  2017, 10(2): 161-190. doi: 10.3934/dcdss.2017009

Spectral notions of aperiodic order

1. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131,33501 Bielefeld, Germany

2. 

Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07743 Jena, Germany

Received  January 2016 Revised  November 2016 Published  January 2017

Various spectral notions have been employed to grasp the structure and the long-range order of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and present generalisations in the form of further examples and remarks.

Citation: Michael Baake, Daniel Lenz. Spectral notions of aperiodic order. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 161-190. doi: 10.3934/dcdss.2017009
References:
[1]

L. Argabright and J. Gil de Lamadrid, Fourier analysis of unbounded measures on locally compact Abelian groups, Memoirs Amer. Math. Soc. , 1 (1974), no. 145.

[2]

J.-B. Aujogue, On embedding of repetitive Meyer multiple sets into model multiple sets, Ergodic Th. & Dynam. Syst., 36 (2016), 1679-1702. doi: 10.1017/etds.2014.133.

[3]

J.-B. AujogueM. BargeJ. Kellendonk and D. Lenz, Equicontinuous factors, proximality and Ellis semigroup for Delone sets, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savien), Birkhäuser, Basel, (2015), pp. 137-194. doi: 10.1007/978-3-0348-0903-0_5.

[4]

J. Auslander, Minimal Flows and their Extensions North-Holland, Amsterdam, 1988.

[5]

M. Baake and F. Gähler, Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples, Top. Appl., 205 (2016), 4-27. doi: 10.1016/j.topol.2016.01.017.

[6]

M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation Cambridge Univ. Press, Cambridge, 2013. doi: 10.1017/CBO9781139025256.

[7]

M. Baake and U. Grimm, Squirals and beyond: Substitution tilings with singular continuous spectrum, Ergodic Th. & Dynam. Syst., 34 (2014), 1077-1102. doi: 10.1017/etds.2012.191.

[8]

M. Baake, C. Huck and N. Strungaru, On weak model sets of extremal density Indag. Math. in press; arXiv: 1512.07129. doi: 10.1016/j.indag.2016.11.002.

[9]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Th. & Dynam. Syst., 24 (2004), 1867-1893. doi: 10.1017/S0143385704000318.

[10]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and topological conjugacy, J. Fourier Anal. Appl., 11 (2005), 125-150. doi: 10.1007/s00041-005-4021-1.

[11]

M. BaakeD. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Th. & Dynam. Syst., 27 (2007), 341-382. doi: 10.1017/S0143385706000800.

[12]

M. BaakeD. Lenz and A. C. D. van Enter, Dynamical versus diffraction spectrum for structures with finite local complexity, Ergodic Th. & Dynam. Syst., 35 (2015), 2017-2043. doi: 10.1017/etds.2014.28.

[13]

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

[14]

M. Baake and R. V. Moody (eds), Directions in Mathematical Quasicrystals CRM Monograph Series, vol. 13, Amer. Math. Soc. , Providence, RI, 2000.

[15]

P. Bak, Icosahedral crystals: Where are the atoms?, Phys. Rev. Lett., 56 (1986), 861-864.

[16]

S. Beckus, D. Lenz, F. Pogorzelski and M. Schmidt, Diffraction theory for processes of tempered distributions, in preparation.

[17]

C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups Springer, Berlin, 1975.

[18]

E. Bombieri and J. E. Taylor, Which distributions of matter diffract? An initial investigation, J. Phys. Colloques, 47 (1986), C3-19-C3-28.

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E. Bombieri and J. E. Taylor, Quasicrystals, tilings and algebraic numbers, Contemp. Math., 64 (1987), 241-264. doi: 10.1090/conm/064/881466.

[20]

X. BressaudF. Durand and A. Maass, Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. London Math. Soc., 72 (2005), 799-816. doi: 10.1112/S0024610705006800.

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X. Deng and R. V. Moody, Dworkin's argument revisited: Point processes, dynamics, diffraction, and correlations, J. Geom. Phys., 58 (2008), 506-541. doi: 10.1016/j.geomphys.2007.12.006.

[26]

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

[27]

M. Einsiedler and T. Ward, Ergodic Theory with a View towards Number Theory GTM 259, Springer, London, 2011. doi: 10.1007/978-0-85729-021-2.

[28]

A. C. D. van Enter and J. Miękisz, How should one define a (weak) crystal?, J. Stat. Phys., 66 (1992), 1147-1153. doi: 10.1007/BF01055722.

[29]

N. P. Frank, Multidimensional constant-length substitution sequences, Top. Appl., 152 (2005), 44-69. doi: 10.1016/j.topol.2004.08.014.

[30]

F. Gähler and R. Klitzing, The diffraction pattern of self-similar tilings, in The Mathematics of Long-Range Aperiodic Order (ed. R. V. Moody), NATO ASI Ser. C 489, Kluwer, Dordrecht, (1997), pp. 141-174.

[31]

J. Gil de Lamadrid and L. N. Argabright, Almost periodic measures Memoirs Amer. Math. Soc. 85 (1990), no. 428. doi: 10.1090/memo/0428.

[32]

J.-B. Gouéré, Diffraction et mesure de Palm des processus ponctuels (Diffraction and Palm measure of point prosesses), C. R. Math. Acad. Sci. Paris, Ser. Ⅰ, 336 (2003), 57-62. doi: 10.1016/S1631-073X(02)00029-8.

[33]

J.-B. Gouéré, Quasicrystals and almost periodicity, Commun. Math. Phys., 255 (2005), 651-681. doi: 10.1007/s00220-004-1271-8.

[34]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. Math., 43 (1942), 332-350. doi: 10.2307/1968872.

[35]

H. Helson, Cocycles on the circle, J. Operator Th., 16 (1986), 189-199.

[36]

A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys., 169 (1995), 25-43. doi: 10.1007/BF02101595.

[37]

Y. Katznelson, An Introduction to Harmonic Analysis 3rd ed. , Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9781139165372.

[38]

J. Kellendonk, Topological Bragg peaks and how they characterise point sets, Acta Phys. Pol. A, 126 (2014), 497-500. doi: 10.12693/APhysPolA.126.497.

[39]

J. Kellendonk and D. Lenz, Equicontinuous Delone dynamical systems, Can. J. Math., 65 (2013), 149-170. doi: 10.4153/CJM-2011-090-3.

[40]

J. Kellendonk and L. Sadun, Meyer sets, topological eigenvalues, and Cantor fiber bundles, J. London Math. Soc., 89 (2013), 114-130. doi: 10.1112/jlms/jdt062.

[41]

B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. USA, 17 (1931), 315-318. doi: 10.1073/pnas.17.5.315.

[42]

J. C. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discr. Comput. Geom., 21 (1999), 161-191. doi: 10.1007/PL00009413.

[43]

J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, vol. 13, Amer. Math. Soc., Providence, RI, (2000), pp. 61-93.

[44]

J.-Y. LeeR. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018. doi: 10.1007/s00023-002-8646-1.

[45]

D. Lenz, Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys., 287 (2009), 225-258. doi: 10.1007/s00220-008-0594-2.

[46]

D. Lenz and R. V. Moody, Stationary processes and pure point diffraction Ergodic Th. & Dynam. Syst. in press; arXiv: 1111.3617. doi: 10.1017/etds.2016.12.

[47]

D. Lenz and C. Richard, Pure point diffraction and cut and project schemes for measures: The smooth case, Math. Z., 256 (2007), 347-378. doi: 10.1007/s00209-006-0077-0.

[48]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in Operator Algebras and Mathematical Physics (eds. J.-M. Combes, J. Cuntz, G.A. Elliott, G. Nenciu, H. Siedentop and S. Stratila), Theta, Bucharest, (2003), pp. 267-285.

[49]

D. Lenz and N. Strungaru, Pure point spectrum for measure dynamical systems on locally compact Abelian groups, J. Math. Pures Appl., 92 (2009), 323-341. doi: 10.1016/j.matpur.2009.05.013.

[50]

L. H. Loomis, An Introduction to Abstract Harmonic Analysis reprint, Dover, New York, 2011.

[51]

Y. Meyer, Nombres de Pisot, Nombres de Salem et Analyse Harmonique LNM 117, Springer, Berlin, 1970.

[52]

Y. Meyer, Algebraic Number Theory and Harmonic Analysis North-Holland, Amsterdam, 1972.

[53]

Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, African Diaspora J. Math., 13 (2012), 1-45.

[54]

Y. Meyer, Measures with locally finite support and spectrum, PNAS, 113 (2016), 3152-3158. doi: 10.1073/pnas.1600685113.

[55]

R. V. Moody, Meyer sets and their duals, in The Mathematics of Long-Range Aperiodic Order (ed. R. V. Moody), NATO ASI Ser. C 489, Kluwer, Dordrecht, (1997), pp. 403-441.

[56]

R. V. Moody, Model sets: A survey, in From Quasicrystals to More Complex Systems (eds. F. Axel, F. D´enoyer and J.P. Gazeau), Springer, Berlin, and EDP Sciences, Les Ulis, (2000), pp. 145-166.

[57]

P. Müller and C. Richard, Ergodic properties of randomly coloured point sets, Can. J. Math., 65 (2013), 349-402. doi: 10.4153/CJM-2012-009-7.

[58]

R. P. Phelps, Lectures on Choquet's Theorem 2nd ed. , LMN 1757, Springer, Berlin, 2001. doi: 10.1007/b76887.

[59]

M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, 2nd ed., LNM 1294, Springer, Berlin, (2010). doi: 10.1007/978-3-642-11212-6.

[60]

E. A. Robinson, On uniform convergence in the Wiener-Wintner theorem, J. London Math. Soc., 49 (1994), 493-501. doi: 10.1112/jlms/49.3.493.

[61]

E. A. Robinson, Symbolic dynamics and tilings of $\mathbb{R}^{d}$, Proc. Sympos. Appl. Math., 60 (2004), 81-119.

[62]

E. A. Robinson, A Halmos-von Neumann theorem for model set dynamical systems, in Dynamics, Ergodic Theory, and Geometry (ed. B. Hasselblatt), MSRI Publications, vol. 54, Cambridge Univ. Press, Cambridge, (2007), pp. 243-272.

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W. Rudin, Fourier Analysis on Groups Wiley, New York, 1962.

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D. ShechtmanI. BlechD. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., 53 (1984), 1951-1953.

[65]

M. Schlottmann, Generalised model sets and dynamical systems, in Directions in mathematical quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, vol. 13, Amer. Math. Soc., Providence, RI, (2000), pp. 143-159.

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B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Syst. , 17 (1997), 695–738 and Ergod. Th. & Dynam. Syst. 19 (1999), 1685 (erratum). doi: 10.1017/S0143385797084988.

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N. Strungaru, On the Bragg diffraction spectra of a Meyer set, Can. J. Math., 65 (2013), 675-701. doi: 10.4153/CJM-2012-032-1.

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N. Strungaru, On weighted Dirac combs supported inside model sets J. Phys. A 47 (2014), 335202 (19 pp); arXiv: 1309.7947. doi: 10.1088/1751-8113/47/33/335202.

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N. Strungaru and V. Terauds, Diffraction theory and almost periodic distributions, J. Stat. Phys., 164 (2016), 1183-1216. doi: 10.1007/s10955-016-1579-8.

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show all references

References:
[1]

L. Argabright and J. Gil de Lamadrid, Fourier analysis of unbounded measures on locally compact Abelian groups, Memoirs Amer. Math. Soc. , 1 (1974), no. 145.

[2]

J.-B. Aujogue, On embedding of repetitive Meyer multiple sets into model multiple sets, Ergodic Th. & Dynam. Syst., 36 (2016), 1679-1702. doi: 10.1017/etds.2014.133.

[3]

J.-B. AujogueM. BargeJ. Kellendonk and D. Lenz, Equicontinuous factors, proximality and Ellis semigroup for Delone sets, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savien), Birkhäuser, Basel, (2015), pp. 137-194. doi: 10.1007/978-3-0348-0903-0_5.

[4]

J. Auslander, Minimal Flows and their Extensions North-Holland, Amsterdam, 1988.

[5]

M. Baake and F. Gähler, Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples, Top. Appl., 205 (2016), 4-27. doi: 10.1016/j.topol.2016.01.017.

[6]

M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation Cambridge Univ. Press, Cambridge, 2013. doi: 10.1017/CBO9781139025256.

[7]

M. Baake and U. Grimm, Squirals and beyond: Substitution tilings with singular continuous spectrum, Ergodic Th. & Dynam. Syst., 34 (2014), 1077-1102. doi: 10.1017/etds.2012.191.

[8]

M. Baake, C. Huck and N. Strungaru, On weak model sets of extremal density Indag. Math. in press; arXiv: 1512.07129. doi: 10.1016/j.indag.2016.11.002.

[9]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Th. & Dynam. Syst., 24 (2004), 1867-1893. doi: 10.1017/S0143385704000318.

[10]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and topological conjugacy, J. Fourier Anal. Appl., 11 (2005), 125-150. doi: 10.1007/s00041-005-4021-1.

[11]

M. BaakeD. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Th. & Dynam. Syst., 27 (2007), 341-382. doi: 10.1017/S0143385706000800.

[12]

M. BaakeD. Lenz and A. C. D. van Enter, Dynamical versus diffraction spectrum for structures with finite local complexity, Ergodic Th. & Dynam. Syst., 35 (2015), 2017-2043. doi: 10.1017/etds.2014.28.

[13]

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

[14]

M. Baake and R. V. Moody (eds), Directions in Mathematical Quasicrystals CRM Monograph Series, vol. 13, Amer. Math. Soc. , Providence, RI, 2000.

[15]

P. Bak, Icosahedral crystals: Where are the atoms?, Phys. Rev. Lett., 56 (1986), 861-864.

[16]

S. Beckus, D. Lenz, F. Pogorzelski and M. Schmidt, Diffraction theory for processes of tempered distributions, in preparation.

[17]

C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups Springer, Berlin, 1975.

[18]

E. Bombieri and J. E. Taylor, Which distributions of matter diffract? An initial investigation, J. Phys. Colloques, 47 (1986), C3-19-C3-28.

[19]

E. Bombieri and J. E. Taylor, Quasicrystals, tilings and algebraic numbers, Contemp. Math., 64 (1987), 241-264. doi: 10.1090/conm/064/881466.

[20]

X. BressaudF. Durand and A. Maass, Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. London Math. Soc., 72 (2005), 799-816. doi: 10.1112/S0024610705006800.

[21]

C. Corduneanu, Almost Periodic Functions 2nd English ed. , Chelsea, New York, 1989.

[22]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory Springer, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[23]

J. M. Cowley, Diffraction Physics 3rd ed. , North-Holland, Amsterdam, 1995.

[24]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes Springer, New York, 1988.

[25]

X. Deng and R. V. Moody, Dworkin's argument revisited: Point processes, dynamics, diffraction, and correlations, J. Geom. Phys., 58 (2008), 506-541. doi: 10.1016/j.geomphys.2007.12.006.

[26]

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

[27]

M. Einsiedler and T. Ward, Ergodic Theory with a View towards Number Theory GTM 259, Springer, London, 2011. doi: 10.1007/978-0-85729-021-2.

[28]

A. C. D. van Enter and J. Miękisz, How should one define a (weak) crystal?, J. Stat. Phys., 66 (1992), 1147-1153. doi: 10.1007/BF01055722.

[29]

N. P. Frank, Multidimensional constant-length substitution sequences, Top. Appl., 152 (2005), 44-69. doi: 10.1016/j.topol.2004.08.014.

[30]

F. Gähler and R. Klitzing, The diffraction pattern of self-similar tilings, in The Mathematics of Long-Range Aperiodic Order (ed. R. V. Moody), NATO ASI Ser. C 489, Kluwer, Dordrecht, (1997), pp. 141-174.

[31]

J. Gil de Lamadrid and L. N. Argabright, Almost periodic measures Memoirs Amer. Math. Soc. 85 (1990), no. 428. doi: 10.1090/memo/0428.

[32]

J.-B. Gouéré, Diffraction et mesure de Palm des processus ponctuels (Diffraction and Palm measure of point prosesses), C. R. Math. Acad. Sci. Paris, Ser. Ⅰ, 336 (2003), 57-62. doi: 10.1016/S1631-073X(02)00029-8.

[33]

J.-B. Gouéré, Quasicrystals and almost periodicity, Commun. Math. Phys., 255 (2005), 651-681. doi: 10.1007/s00220-004-1271-8.

[34]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. Math., 43 (1942), 332-350. doi: 10.2307/1968872.

[35]

H. Helson, Cocycles on the circle, J. Operator Th., 16 (1986), 189-199.

[36]

A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys., 169 (1995), 25-43. doi: 10.1007/BF02101595.

[37]

Y. Katznelson, An Introduction to Harmonic Analysis 3rd ed. , Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9781139165372.

[38]

J. Kellendonk, Topological Bragg peaks and how they characterise point sets, Acta Phys. Pol. A, 126 (2014), 497-500. doi: 10.12693/APhysPolA.126.497.

[39]

J. Kellendonk and D. Lenz, Equicontinuous Delone dynamical systems, Can. J. Math., 65 (2013), 149-170. doi: 10.4153/CJM-2011-090-3.

[40]

J. Kellendonk and L. Sadun, Meyer sets, topological eigenvalues, and Cantor fiber bundles, J. London Math. Soc., 89 (2013), 114-130. doi: 10.1112/jlms/jdt062.

[41]

B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. USA, 17 (1931), 315-318. doi: 10.1073/pnas.17.5.315.

[42]

J. C. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discr. Comput. Geom., 21 (1999), 161-191. doi: 10.1007/PL00009413.

[43]

J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, vol. 13, Amer. Math. Soc., Providence, RI, (2000), pp. 61-93.

[44]

J.-Y. LeeR. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018. doi: 10.1007/s00023-002-8646-1.

[45]

D. Lenz, Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys., 287 (2009), 225-258. doi: 10.1007/s00220-008-0594-2.

[46]

D. Lenz and R. V. Moody, Stationary processes and pure point diffraction Ergodic Th. & Dynam. Syst. in press; arXiv: 1111.3617. doi: 10.1017/etds.2016.12.

[47]

D. Lenz and C. Richard, Pure point diffraction and cut and project schemes for measures: The smooth case, Math. Z., 256 (2007), 347-378. doi: 10.1007/s00209-006-0077-0.

[48]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in Operator Algebras and Mathematical Physics (eds. J.-M. Combes, J. Cuntz, G.A. Elliott, G. Nenciu, H. Siedentop and S. Stratila), Theta, Bucharest, (2003), pp. 267-285.

[49]

D. Lenz and N. Strungaru, Pure point spectrum for measure dynamical systems on locally compact Abelian groups, J. Math. Pures Appl., 92 (2009), 323-341. doi: 10.1016/j.matpur.2009.05.013.

[50]

L. H. Loomis, An Introduction to Abstract Harmonic Analysis reprint, Dover, New York, 2011.

[51]

Y. Meyer, Nombres de Pisot, Nombres de Salem et Analyse Harmonique LNM 117, Springer, Berlin, 1970.

[52]

Y. Meyer, Algebraic Number Theory and Harmonic Analysis North-Holland, Amsterdam, 1972.

[53]

Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, African Diaspora J. Math., 13 (2012), 1-45.

[54]

Y. Meyer, Measures with locally finite support and spectrum, PNAS, 113 (2016), 3152-3158. doi: 10.1073/pnas.1600685113.

[55]

R. V. Moody, Meyer sets and their duals, in The Mathematics of Long-Range Aperiodic Order (ed. R. V. Moody), NATO ASI Ser. C 489, Kluwer, Dordrecht, (1997), pp. 403-441.

[56]

R. V. Moody, Model sets: A survey, in From Quasicrystals to More Complex Systems (eds. F. Axel, F. D´enoyer and J.P. Gazeau), Springer, Berlin, and EDP Sciences, Les Ulis, (2000), pp. 145-166.

[57]

P. Müller and C. Richard, Ergodic properties of randomly coloured point sets, Can. J. Math., 65 (2013), 349-402. doi: 10.4153/CJM-2012-009-7.

[58]

R. P. Phelps, Lectures on Choquet's Theorem 2nd ed. , LMN 1757, Springer, Berlin, 2001. doi: 10.1007/b76887.

[59]

M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, 2nd ed., LNM 1294, Springer, Berlin, (2010). doi: 10.1007/978-3-642-11212-6.

[60]

E. A. Robinson, On uniform convergence in the Wiener-Wintner theorem, J. London Math. Soc., 49 (1994), 493-501. doi: 10.1112/jlms/49.3.493.

[61]

E. A. Robinson, Symbolic dynamics and tilings of $\mathbb{R}^{d}$, Proc. Sympos. Appl. Math., 60 (2004), 81-119.

[62]

E. A. Robinson, A Halmos-von Neumann theorem for model set dynamical systems, in Dynamics, Ergodic Theory, and Geometry (ed. B. Hasselblatt), MSRI Publications, vol. 54, Cambridge Univ. Press, Cambridge, (2007), pp. 243-272.

[63]

W. Rudin, Fourier Analysis on Groups Wiley, New York, 1962.

[64]

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Figure 1.  A central patch of the eightfold symmetric Ammann-Beenker tiling of the plane, which can be generated by an inflation rule and is thus a self-similar tiling; see [6,Sec. 6.1] for details. The set of its vertex points is an example of a Meyer set, hence it is also an FLC Delone set. Moreover, it is a regular model set, as described in detail in [6,Ex. 7.8].
Figure 2.  Illustration of a central patch of the diffraction measure of the Ammann-Beenker point set of Figure 1, which has pure point diffraction. A Bragg peak of intensity $I$ at $k\in\mathcal{B}$ is represented by a disc of an area that is proportional to $I$ and centred at $k$. Here, $\mathcal{B}$ is a scaled version of $\mathbb{Z}[{\rm{e}}^{\pi {\rm{i}}/4}]$, which is a group; compare [6,Sec. 9.4.2] for details. Although $\mathcal{B}$ is dense, the figure only shows Bragg peaks beyond a certain threshold. In particular, there are no extinctions in this case. At the same time, this measure is the diffraction measure of the Delone dynamical system defined by the (strictly ergodic) hull of the Ammann-Beenker point set, and $\mathcal{B}$ is its dynamical spectrum for the translation action of $\mathbb{R}^{2}$ on the hull.
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