Deformation of delone dynamical systems and pure point diffraction

Journal of Fourier Analysis and Applications

Volumn 11, Issue 2, 2005, Pages 125-150

  Baake, Michael ^a   Lenz, Daniel ^b  

^a BIELEFELD UNIVERSITY   (Germany)

^b CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 16244414297     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-005-4021-1     Document Type: Article

References (52)

1. Baake, M. (2002). Diffraction of weighted lattice subsets, Canad. Math. Bull. 45, 483-498; math.MG/0106111.
2. Baake, M. and Höffe, M. (2000). Diffraction of random tilings: Some rigorous results, J. Statist. Phys. 99, 219-261; math-ph/9904005.
3. Baake, M. and Lenz, D. (2004). Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems 24, 1867-1893; math.DS/0302061.
4. Baake, M., Lenz, D., and Moody, R. V. A characterization of model sets by dynamical systems, in preparation.
5. Baake, M. and Moody, R. V. (1998). Diffractive point sets with entropy, J. Phys. A: Math. Gen. 31, 9023-9039; math-ph/9809002.
6. Baake, M. and Moody, R. V. Self-similar measures for quasicrystals, in: [7], pp. 1-42; math.MG/0008063.
7. Baake, M. and Moody, R. V., Eds., (2000). Directions in mathematical quasicrystals, CRM Monograph Series, Vol. 13, AMS, Rhode Island.
8. Baake, M. and Moody, R. V. (2004). Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. (Crelle), 573, 61-94; math.MG/0203030.
9. Bak, P. (1986). Icosahedral crystals from cuts in six-dimensional space, Scripts Met. 20, 1199-1204.
10. Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups, Springer, Berlin.
11. Bernuau, G. and Duneau, M. Fourier analysis of deformed model sets, in: [7], pp. 43-60.
12. Clark, A. and Sadun, L. (2003). When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems 23, 1043-1057; math.DS/0201152.
13. Clark, A. and Sadun, L. (2004). When shape matters: Deformations of tiling spaces, Ergodic Theory Dynam. Systems in press; math.DS/0306214.
14. Cowley, J. M. (1995). Diffraction Physics, 3rd ed., North-Holland, Amsterdam.
15. Denker, M., Grillenberger, C., and Sigmund, K. (1976). Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527, Springer, Berlin.
16. Dworkin, S. (1993). Spectral theory and X-ray diffraction, J. Math. Phys. 34,2965-2967.
17. Fisher, B. N. (2004). Private communication.
18. Fisher, B. N. and Rabson, D. A. (2004). Group cohomology and quasicrystals, I: Classification of two-dimensional space groups, Ferroelectrics 305, 37-40; and II: The three crystallographic invariants in two and three dimensions, Ferroelectrics 305, 25-28, (2004).
19. Fisher, B. N. and Rabson, D. A. (2004). Group cohomology and quasicrystals, I: Classification of two-dimensional space groups, Ferroelectrics 305, 37-40; and II: The three crystallographic invariants in two and three dimensions, Ferroelectrics 305, 25-28, (2004).
20. Furstenberg, M. (1981). Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, NJ.
21. Gil de Lamadrid, J. and Argabright, L.N. (1990). Almost periodic measures, Mem. Amer. Math. Soc. 85(428), AMS, Providence, RI.
22. Gouéré, J.-B. (2003). Diffraction et mesure de palm des processus ponctuels, C. R. Math. Acad. Sci. Paris, Sel. 1336 57-62; math.PR/0208064.
23. Gouéré, J.-B. (2004). Quasicrystals and almost periodicity, Comm. Math. Phys. in press; math-ph/0212012.
24. Hof, A. (1995). On diffraction by aperiodic structures, Comm. Math. Phys. 169, 25-43.
25. Hof, A. Diffraction by aperiodic structures, in: [34], pp. 239-268.
26. Hof, A. Uniform distribution and the projection method, in: [38], pp. 201-206.
27. Hof, A. (1995). Diffraction by aperiodic structures at high temperatures, J. Phys. A: Math. Gen. 28, 57-62.
28. Kelley, J. L. (1955). General Topology, van Nostrand, Princeton, NJ; reprint, Springer, New York, (1975).
29. Krengel, U. (1985). Ergodic Theorems, de Gruyter, Berlin.
30. Lagarias, J. C. (1999). Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom. 21, 345-372.
31. Lee, J.-Y., Moody, R. V., and Solomyak, B. (2002). Pure point dynamical and diffraction spectra, Ann. Henri Poincaré 3, 1003-1018; mp_arc/02-39.
32. Lee, J.-Y., Moody, R. V., and Solomyak, B. (2003). Consequences of pure point diffraction spectra for multiset substitution systems, Discrete Comput. Geom. 29, 525-560.
33. Loomis, L. H. (1953). An Introduction to Abstract Harmonic Analysis, van Nostrand, Princeton, NJ.
34. Moody, R. V. Meyer sets and their duals, in: [34], pp. 403-441.
35. Moody, R. V., Ed., (1997). The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C 489, Kluwer, Dordrecht.
36. Moody, R. V. (2000). Model sets: A survey, in From Quasicrystals to More Complex Systems, Axel, F., Dénoyer, F., and Gazeau, J.P., Eds., EDP Sciences, Les Ulis, and Springer, Berlin, 145-166; math.MG/0002020.
37. Moody, R. V. (2002). Uniform distribution in model sets, Canad. Math. Bull. 45, 123-130.
38. Nielsen, O. A. (1980). Direct Integral Theory, Marcel-Dekker, New York.
39. Patera, J., Ed., (1998). Quasicrystals and Discrete Geometry, Fields Institute Monographs, Vol. 10, AMS, Providence, RI.
40. Pedersen, G. K. (1989). Analysis Now, Springer, New York; rev. printing (1995).
41. Robinson, E. A. (1994). On uniform convergence in the Wiener-Wintner Theorem, J. London Math. Soc. (2) 49, 493-501.
42. Rudin, W. (1962). Fourier Analysis on Groups, Wiley, New York; reprint (1990).
43. Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translation symmetry, Phys. Rev. Lett. 53, 183-185.
44. Schlottmann, M. Cut-and-project sets in locally compact Abelian groups, in: [38], pp. 247-264.
45. Schlottmann, M. Generalized model sets and dynamical systems, in: [7], pp. 143-159.
46. Solomyak, B. Spectrum of dynamical systems arising from Delone sets, in: [38], pp. 265-275.
47. Solomyak, B. (1997). Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17, 695-738; Erratum: Ergodic Theory Dynam. Systems 19, p. 1685, (1999).
48. Suck, J.-B., Schreiber, M., and Häussler, P., Eds., (2002). Quasicrystals - An Introduction to Structure, Physical Properties, and Applications, Springer, Berlin.
49. Tempelman, A. (1992). Ergodic Theorems for Group Actions, Kluwer, Dordrecht.
50. Trebin, H.-R., Ed., (2003). Quasicrystals-Structure and Physical Properties, Wiley-VCH, Weinheim.
51. Walters, P. (1982). An Introduction to Ergodic Theory, Springer-Verlag, New York.
52. Weidmann, J. (1980). Linear Operators in Hilbert Spaces, Springer-Verlag, New York