×

Pure discrete spectrum dynamical system and periodic tiling associated with a substitution. (English) Zbl 1083.37009

This survey studies symbolic dynamical systems associated with a substitution of Pisot type. The goal is to find a computable sufficient condition in order to have a pure discrete spectrum. The author uses a formal and algebraic point of view instead of the usual measurable techniques to investigate the questions of self-similarity and pure discrete spectrum.

MSC:

37B10 Symbolic dynamics
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
47A35 Ergodic theory of linear operators
28A80 Fractals
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Pisot substitutions and Rauzy fractals, Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000), 181-207 (2001) · Zbl 1007.37001
[2] Les nombres \(p\)-adiques, no 14 (1975) · Zbl 0313.12104
[3] Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, 130, 619-626 (2002) · Zbl 1028.37008
[4] Which distributions of matter diffract? An initial investigation, International workshop on aperiodic crystals (Les Houches, 1986), 47, C3-19-C3-28 (1986) · Zbl 0693.52002
[5] Connectedness of geometric representation of substitutions of Pisot type · Zbl 1031.37015
[6] Automate des préfixes-suffixes associé à une substitution primitive, J. Théor. Nombres Bordeaux, 13, 353-369 (2001) · Zbl 1071.37011
[7] Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc., 353, 5121-5144 (2001) · Zbl 1142.37302
[8] The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41, 221-239 (197778) · Zbl 0348.54034
[9] Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci., 65, 153-169 (1989) · Zbl 0679.10010
[10] Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems, 20, 1061-1078 (2000) · Zbl 0965.37013
[11] Corrigendum and addendum to: “Linearly recurrent subshifts have a finite number of non-periodic subshift factors”, Ergodic Theory Dynam. Systems, 23, 663-669 (2003)
[12] Substitution dynamical systems: algebraic characterization of eigenvalues, Ann. Sci. École Norm. Sup., 29, 4, 519-533 (1996) · Zbl 0866.11023
[13] An invertible substitution with a non-connected Rauzy fractal (2002)
[14] Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable, Ergodic Theory Dynam. Systems, 6, 4, 529-540 (1986) · Zbl 0625.28011
[15] Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory Dynam. Systems, 23, 533-540 (2003) · Zbl 1031.11010
[16] Directed graphs and substitutions, Theory Comput. Syst., 34, 545-564 (2001) · Zbl 0993.68075
[17] Frontière du fractal de Rauzy et système de numération complexe, Acta Arith., 95, 3, 195-224 (2000) · Zbl 0968.28005
[18] Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci., 99, 2, 327-334 (1992) · Zbl 0763.68049
[19] Substitutions in Dynamics, Arithmetics and Combinatorics, 1794 (2002) · Zbl 1014.11015
[20] Substitution dynamical systems-spectral analysis, 1294 (1987) · Zbl 0642.28013
[21] Nombres algébriques et substitutions, Bull. Soc. Math. France, 110, 2, 147-178 (1982) · Zbl 0522.10032
[22] Rotations sur les groupes, nombres algébriques et substitutions, Séminaire de Théorie des Nombres (Talence, 1987-1988), 21-01-21-12 (1988) · Zbl 0726.11019
[23] Quasicrystals and geometry (1995) · Zbl 0828.52007
[24] Représentation géométrique, combinatoire et arithmétique des substitutions de type Pisot (2000)
[25] Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic Theory Dynam. Systems, 23, 1247-1273 (2003) · Zbl 1052.37009
[26] Geodesic laminations as geometric realizations of Pisot substitutions, Ergodic Theory Dynam. Systems, 20, 1253-1266 (2000) · Zbl 0963.37013
[27] Uniform algebraic approximation of shift and multiplication operators, Dokl. Akad. Nauk SSSR, 259, 3, 526-529; English transl.: Soviet Math. Dokl. 24-1 (1981), 97-10 (1981) · Zbl 0484.47005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.