Quasicrystallic Tilings and Projection Method
Tóm tắt
Basic notions related to quasiperiodic tilings and Delone sets in Eucledean space are discussed. It is shown how the cut and project method of constructing them is used to calculate their spectra. Special attention is paid to self-similar tilings and the way one can obtain one-dimensional substitutional tilings by the projection scheme. Bibliography: 18 titles.
Tài liệu tham khảo
P. Arnoux and S. Ito, “Pisot substitutions and Rauzy fractals,” Preprint No. 98-18, Institut de Mathématiques de Luminy (1998).
M. Baake, R. V. Moody, and M. Schlottmann, “Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces,” J. Phys. A, 31,No. 27, 5755–5765 (1998).
M. Baake and R. V. Moody, “Self-similar measures for quasicrystals,” in: Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Providence (2000), pp. 1–42.
M. Baake, “A Guide to Mathematical Quasicrystals,” arXiv: math-ph/9901014.
N. G. de Bruijn, “Algebraic theory of Penrose's non-periodic tilings of the plane. I, II,” Kon. Nederl. Akad. Wetensch., A84, 39–66 (1981).
N. G. de Bruijn, “Quasicrystals and their Fourier transform,” Kon. Nederl. Akad. Wetensch., Proc. Ser., A89, 123–152 (1986).
F. M. Dekking, “The spectrum of dynamical systems arising from substitutions of constant length,” Zeit. Wahr., 41, 221–239 (1978).
A. Hof, Quasicrystals, Aperiodicity and Lattice Systems, Thesis, Rijksununiversiteit Groningen (1992).
J. E. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math. J., 30, 713–743 (1981).
Y. Meyer, “Algebraic numbers and harmonic analysis,” in: North-Holland Mathematical Library, Vol. 2, North-Holland, New York (1972).
Y. Meyer “Quasicrystals, Diophantine approximation and algebraic numbers,” in: Beyond Quasicrystals, Les Houches, Springer, Berlin (1995), pp. 3–16.
R. Penrose, “The role of aesthetics in pure and applied mathematical research,” Bull. Inst. Math. Appl., 10, 266–271 (1974).
B. Solomyak, “Dynamics of Self-Similar Tiling,” Ergod. Th. & Dyn. Syst., 17, 695–738 (1997).
B. Solomyak, “Spectrum of dynamical systems arising from Delone sets,” in: Quasicrystals and Discrete Geometry, (Toronto, ON, 1995), Fields Inst. Monogr., 10, Amer. Math. Soc., Providence (1998), pp. 265–275.
D. Schechtman, I. Blech, D. Gratias, and J. Cahn “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett., 53, 1951–1953 (1984).
A. M. Vershik and A. N. Livshits, “Adic models of ergodic transformations, spectral theory, substitutions, and related topics,” Adv. Sov. Math., 9, 185–204 (1992).
D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie, Springer-Verlag (1996).
E. Yu. Shevkoplyas, Diffraction spectra of quasicrystals, Graduation Thesis, St. Petersburg University (2000).