We survey mathematical properties of quasicrystals, first from the point of view of harmonic analysis, then from the point of view of morphic and automatic sequences.
Nous proposons un tour dʼhorizon des propriétés mathématiques des quasicristaux, dʼabord du point de vue de lʼanalyse harmonique, ensuite du point de vue des suites morphiques et automatiques.
Mot clés : Quasicristaux, Ensembles modèles, Suites automatiques
Jean-Paul Allouche 1; Yves Meyer 2
@article{CRPHYS_2014__15_1_6_0, author = {Jean-Paul Allouche and Yves Meyer}, title = {Quasicrystals, model sets, and automatic sequences}, journal = {Comptes Rendus. Physique}, pages = {6--11}, publisher = {Elsevier}, volume = {15}, number = {1}, year = {2014}, doi = {10.1016/j.crhy.2013.09.002}, language = {en}, }
Jean-Paul Allouche; Yves Meyer. Quasicrystals, model sets, and automatic sequences. Comptes Rendus. Physique, Volume 15 (2014) no. 1, pp. 6-11. doi : 10.1016/j.crhy.2013.09.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2013.09.002/
[1] Nombres de Pisot, nombres de Salem et analyse harmonique, Lect. Notes Math., vol. 117, Springer-Verlag, 1970
[2] Algebraic Numbers and Harmonic Analysis, North-Holland, 1972
[3] Quasicrystals, Diophantine approximation and algebraic numbers (F. Axel; D. Gratias, eds.), Beyond Quasicrystals, Les Éditions de Physique, Springer, 1995, pp. 3-16
[4] Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, Afr. Diaspora J. Math., Volume 13 (2012), pp. 1-45
[5] Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett., Volume 50 (1983), pp. 1870-1872
[6] One-dimensional Schrödinger equation with an almost periodic potential, Phys. Rev. Lett., Volume 50 (1983), pp. 1873-1876
[7] Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom., Volume 21 (1999), pp. 161-191
[8] Quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ., Volume 55 (2010), pp. 947-964
[9] Exponential Riesz bases, discrepancy of irrational rotations and BMO, J. Fourier Anal. Appl., Volume 17 (2011), pp. 879-898
[10] Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., Volume 53 (1984), pp. 1951-1953
[11] Schrödinger difference equation with deterministic ergodic potentials (F. Axel; D. Gratias, eds.), Beyond Quasicrystals, Les Éditions de Physique, Springer, 1995, pp. 481-549
[12] Symbolic dynamics II, Sturmian trajectories, Am. J. Math., Volume 62 (1940), pp. 1-42
[13] Algebraic Combinatorics on Words, Cambridge University Press, 2002
[14] Symbolic dynamics, Am. J. Math., Volume 60 (1938), pp. 815-866
[15] Sequences with minimal block growth, Math. Syst. Theory, Volume 7 (1973), pp. 138-153
[16] The ubiquitous Prouhet–Thue–Morse sequence, Singapore, 1998 (Discrete Math. Theor. Comput. Sci.), Springer, London (1999), pp. 1-16
[17] Ensembles presque périodiques k-reconnaissables, Theor. Comput. Sci., Volume 9 (1979), pp. 141-145
[18] Suites algébriques, automates et substitutions, Bull. Soc. Math. Fr., Volume 108 (1980), pp. 401-419
[19] Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003
[20] Automata and automatic sequences (F. Axel; D. Gratias, eds.), Beyond Quasicrystals, Les Éditions de Physique, Springer, 1995, pp. 293-367
[21] On Sturmian sequences which are invariant under some substitutions, Kyoto, 1997, Kluwer Academic Publishers (1999), pp. 347-373
[22] On substitution invariant Sturmian words: an application of Rauzy fractals, RAIRO Theor. Inform. Appl., Volume 41 (2007), pp. 329-349
[23] Nombres algébriques et substitutions, Bull. Soc. Math. Fr., Volume 110 (1982), pp. 147-178
[24] Repetitive Delone sets and quasicrystals, Ergod. Theor. Dyn. Syst., Volume 23 (2003), pp. 831-867
[25] Automatic sets and Delone sets, J. Phys. A, Math. Gen., Volume 37 (2004), pp. 4017-4038
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