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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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. Google Scholar

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

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

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

[24]

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

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

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

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

[52]

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[53]

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

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

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[23]

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

[24]

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

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

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

[51]

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

[52]

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

[53]

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

[54]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[58]

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

[59]

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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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