Summary
Applications of the theory of finite automata, automatic and substitutive sequences to the description of physical situations intermediate between crystallographic order and random disorder in theoretical and experimental Condensed Matter Physics are described. Particular reference is made to Trace Mapping techniques, experimental applications include the investigation of multilayer heterostructures designed after such automatic or substitutive sequences and the use of their specific properties.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
F.Abeles, Recherche sur la propagation des ondes electromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces, Annales de Physique vol.5 (1950), 596–640 and 707–782.
J.-P.Allouche, J.Peyrière, Sur une formule de récurrence sur les traces de produits de matrices associés certaines substitutions, C.R.Acad.Sc.Paris, t.302, Série II, (1986)18, 1135.
Y.Avishai, D.Berend, Trace maps for arbitrary substitution sequences, J.Phys.A Math.Gen.26 (1993) 2437–2443.
F.Axel, J.P.Allouche, M.Kléman, M.Mendès France, J.Peyrière, Vibrational modes in a one dimensional “quasi alloy”: the Morse case, J. Phys. (Paris) 47, C3 (1986) 181–186.
F.Axel, J.P.Allouche, Z.Y.Wen, On certain properties of high resolution X ray diffraction spectra of finite size generalized Rudin-Shapiro multilayer heterostructures, J. Phys.Cond.Matter.4 (1992) 8713–8728.
F.Axel, F.Dénoyer, J.P.Gazeau, “From Quasicrystals to more complex systems,” Courses of the Winter School Order, Chance and Risk, F.Axel, F.Dénoyer, and J.P.Gazeau, Editors, 374 p., with a glossary, Les Editions de Physique/Springer-Verlag, (2000).
F.Axel, D.Gratias, “Beyond Quasicrystals”, Courses of the Winter School “Beyond Quasicrystals”, F.Axel and D.Gratias, Editors, with a glossary, 630p., Les Editions de Physique/Springer-Verlag, (1995).
F.Axel, J.Peyrière, Etats étendus dans une chaîne à désordre contrôlé, C.R.Acad.Sc.Paris, t. 306, Série II, (1988), 179–182.
F.Axel, J.Peyrière, Spectrum and extended states in a harmonic chain with controlled disorder: effects of the Thue-Morse symmetry, J.Stat.Phys.57 (1989) 1013–1047.
F.Axel, H.Terauchi, High resolution X ray diffraction spectra of Thue-Morse GaAs-AlAs heterostructures: towards a novel description of disorder, Phys.Rev.Lett.66 (1991) 2223 and Reply Phys.Rev.Lett.73 (1994) 1308–1311.
E.Bombieri, J.Taylor, Which distributions of matter diffract? J. Phys., C3, 47 (1986) 19.
Z.Cheng, R.Savit, R.Merlin, Structure and electronic properties of Thue-Morse lattices, Phys.Rev.B37 (1988) 4375.
G.Christol, T.Kamae, M.Mendès France, G.Rauzy, Suites algébriques, automates et substitutions, Bull.Soc.Math.Fr.108 (1980) 401–419 and Course 11 in[7].
L.Dal Negro, C.J.Oton, Z.Gaburro, L.Pavesi, P.Johnson, A.Lagendijk, R.Righini, M.Colocci, D.S.Wiersma, Light transport through the band-edge states of Fibonacci quasicrystals, Phys.Rev.Lett.90 (2003) 55501.
R.Fricke, F.Klein, Vorlesungen über die Theorie des Automorphen Functionen, B.G.Teubner, Leipzig (1897).
W.Gellerman, M.Kohmoto, B.Sutherland, P.C.Taylor, Localization of light waves in Fibonacci dielectric multilayers, Phys.Rev.Lett.72 (1994) 633.
C.Godrèche, J.M.Luck, F.Vallet, Quasiperiodicity and types of order; a study in one dimension, J.Phys.A Math.Gen.20 (1987) 4483–4499.
D.Gratias, L.Michel, “Aperiodic Crystals,” Proceedings of the Workshop in les Houches (March 11–20, 1986) D.Gratias and Louis Michel, Editors, J. Phys. (Paris) 47, C3 (1986).
R.D.Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm.Pure and Applied Math.XXV (1972) 635–649
R.D.Horowitz, Induced automorphism on Fricke characters of free groups, Trans.Am.Math.Soc.208 (1975) 41–50.
A.Hu, Z.X.Wen, S.S.Jiang, W.T.Tong, R.-W.Peng, D.Feng, One-dimensional k-component Fibonacci structures, Phys.Rev.B48 (1993), 829.
M.Kohmoto, L.P.Kadanoff, C.Tang, Localization problem in one dimension: mapping and escape, Phys.Rev.Lett.50 (1983), 1870.
M.Kohmoto, B.Sutherland, K.Iguchi, Localization in optics: quasiperiodic media, Phys.Rev.Lett.58 (1987), 2436.
H.Kunz, Quantized currents and topological invariants for electrons in incommensurate potentials, Phys.Rev.Lett.57 (1986), 1095.
B.Leblanc, E.Lutton, F.Axel, Genetic algorithms as a tool in the study of aperiodic order with application to the case of X-ray diffraction spectra of GaAs-AlAs multilayer heterostructures, Euro.Phys.J.B29 (2002) 619–628.
N.-H.Liu, Propagation of light waves in Thue-Morse dielectric multilayers, Phys.Rev.B55 (1997), 3543.
Y.Liu, R.Riklund, Electronic properties of perfect and non-perfect one-dimensional quasicrystals, Phys.Rev.B35 (1987), 6034.
R.Merlin, K.Bajema, R.Clarke, F.Y.Juang, P.K.Bhattacharya, Quasiperiodic GaAl-AlAs heterostructures, Phys.Rev.Lett.55 (1985), 1768.
R.Merlin, K.Bajema, J.Nagle, K.Ploog, Raman scattering by acoustic phonons and structural properties of Fibonacci, Thue-Morse and random superlattices, J.Phys.(Paris) Colloq 48 (1987) C5 503–506.
L.Moretti, I.Rea, L.Rotiroti, I.Rendina, G.Abbate, A.Marino, L.De Stefano, Photonic band gaps analysis of Thue-Morse multilayers made of porous silicon, Opt. Express 14 (2006), 6264.
M.Morse, Recurrent geodesics on a surface of negative curvature, Trans.Am.Math.Soc.2 (1921) 84–100.
S.Ostlund, R.Pandit, Renormalization-group analysis of the discrete quasiperiodic Schrödinger equation, Phys.Rev.B29 (1984), 1394.
S.Ostlund, R.Pandit.D.Rand, H.-J.Schellnhuber, E.Siggia, One-dimensional Schrödinger equation with an almost periodic potential, Phys.Rev.Lett.50 (1983) 1873.
J.Peyrière, On the trace map for products of matrices associated with substitutive sequences, J.Stat.Phys. 62 (1991) 411–414, see also Course 16 in[7].
J.Peyrière, E.Cockayne, F.Axel, Line-shape analysis of high resolution X-ray diffraction spectra of finite size Thue-Morse GaAs-AlAs multilayer heterostructures, J.Phys. I Fr., 5 (1995), 111–127.
J.Peyrière, Z.-Y.Wen, Z.-X.Wen, Polynômes associés aux endomorphismes de groupes libres, L’Enseignement Mathématique, vol 39, (1993).
E.Prouhet, Mémoire sur quelques relations entre les puissances des nombres, C.R.Acad.Sci.Paris 33 (1851) 225.
Lord Rayleigh, On the propagation of waves through a stratified medium, with special reference to the question of reflection, Proc.Roy.Soc.(1912), 86A, 207–226.
R.Riklund, M.Severin, Optical properties of perfect and non-perfect quasi-periodic multilayers: a comparison with periodic and disordered multilayers, J.Phys.C Solid State Phys.21 (1988), 3217.
C.S.Ryu, G.Y.Oh, M.H.Lee, Electronic properties of a tight-binding and a Kronig-Penney model of the Thue-Morse chain, Phys.Rev.B48 (1993), 132.
D.Shechtman, I.Blech, D.Gratias, J.W.Cahn, Metallic phases with long range orientational order and no translational symmetry, Phys.Rev.Lett.53 (1984) 1951–1953.
H.Terauchi et al., Xray studies of semiconductor superlattices grown by MBE, Xray diffraction pattrens of configurational Fibonacci lattices and Synchrotron Xray diffraction study of third-order Fibonacci lattices, J.Phys.Soc.Jpn.54 (1985) 4576, 57 (1988) 2416, 59 (1990), 405.
A.Thue, Uber unendliche Zeichenreihen, Norske vid.Selsk.Skr.I.Mat.Kl.Christiania 7 (1906) 1–22, 13 (1912) 1–67.
J.Todd, R.Merlin, R.Clarke, K.M.Mohanty, J.D.Axe, Synchrotron X-ray study of a Fibonacci superlattice, Phys.Rev.Lett.57 (1986) 1157.
Z.Wang, R.W.Peng, F.Qiu, Z.H.Tang, X.F.Hu, X.Q.Huang, M.Wang, A.Hu, S.S.Jiang, D.Feng, Structural fabrication and defect-dependent optical transmission of dielectric multilayers, J.Cryst. Growth 275 (2005) 209.
Z.-X.Wen, Relations polynomiales entre les traces de produits de matrices, C.R.Acad.Sc.Paris, t.318, Série I, (1994) 99–104.
Z.-X.Wen, Z.-Y.Wen, On the leading term and the degree of the polynomial trace mapping associated with a substitution, J.Stat.Phys.75 (1994), 627–641.
Z.-Y.Wen, F.Wijnands, J.S.W.Lamb, A natural class of generalized Fibonacci chains, J.Phys.A Math.Gen.27 (1994), 3689.
P.Yeh, A.Yariv, C.-S.Hong, Electromagnetic propagation in stratified media; I General theory; II Birefringence, phase matching and X-ray lasers, J.Opt. Soc.Am.67 (1977), 423–438, 438–448.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Axel, F. (2010). A Crash Look into Applications of Aperiodic Substitutive Sequences. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4888-6_25
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4888-6_25
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4887-9
Online ISBN: 978-0-8176-4888-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)