Scaling of the Thue-Morse Diffraction Measure

• Authors: M. Baake , U. Grimm , J. Nilsson
• Languages of publication: EN

Abstracts

EN

We revisit the well-known and much studied Riesz product representation of the Thue-Morse diffraction measure, which is also the maximal spectral measure for the corresponding dynamical spectrum in the complement of the pure point part. The known scaling relations are summarised, and some new findings are explained.

Keywords

EN

61.05.cc; 61.43.-j; 61.44.Br

Discipline

• 61.43.-j: Disordered solids(see also 81.05.Gc Amorphous semiconductors, 81.05.Kf Glasses, and 81.05.Rm Porous materials; granular materials in materials science; for photoluminescence of disordered solids, see 78.55.Mb and 78.55.Qr)
• 61.44.Br: Quasicrystals
• 61.05.cc: Theories of x-ray diffraction and scattering

• Journal: Acta Physica Polonica A
• Year: 2014
• Volume: 126
• Issue: 2
• Pages: 431-434

Dates

published

2014-08

Contributors

author

M. Baake

• Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

author

U. Grimm

• Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK

author

J. Nilsson

• Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

References

• [1] J.-P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, Cambridge 2003
• [2] M. Baake, U. Grimm, Aperiodic Order, Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge 2013
• [3] M. Keane, Z. Wahrscheinlichkeitsth. verw. Geb. 10, 335 (1968), doi: 10.1007/BF00531855
• [4] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, LNM 1294, 2nd ed., Springer, Berlin 2010
• [5] A.C.D. van Enter, J. Miękisz, J. Stat. Phys. 66, 1147 (1992), doi: 10.1007/BF01055722
• [6] M. Baake, U. Grimm, J. Phys. A: Math. Theor. 41, 422001 (2008), doi: 10.1088/1751-8113/41/42/422001
• [7] A. Zygmund, Trigonometric Series, 3rd ed., Cambridge University Press, Cambridge 2002
• [8] N.P. Frank, Topol. Appl. 152, 44 (2005), doi: 10.1016/j.topol.2004.08.014
• [9] N. Wiener, J. Math. Massachusetts 6, 145 (1927)
• [10] K. Mahler, J. Math. Massachusetts 6, 158 (1927)
• [11] S. Kakutani, in: Proc. 6th Berkeley Symp. on Math. Statistics and Probability, Eds. L.M. LeCam, J. Neyman, E.L. Scott, Univ. of California Press, Berkeley 1972, p. 319
• [12] R.L. Withers, Z. Krist. 220, 1027 (2005), doi: 10.1524/zkri.2005.220.12.1027
• [13] Z. Cheng, R. Savit, R. Merlin, Phys. Rev. B 37, 4375 (1988), doi: 10.1364/JOSAB.29.002130
• [14] C. Godréche, J.M. Luck, J. Phys. A, Math. Gen. 23, 3769 (1990), doi: 10.1088/0305-4470/23/16/024
• [15] M.A. Zaks, J. Phys. A, Math. Gen. 35, 5833 (2002), doi: 10.1088/0305-4470/35/28/304
• [16] M.A. Zaks, A.S. Pikovsky, J. Kurths, J. Stat. Phys. 88, 1387 (1997), doi: 10.1007/BF02732440
• [17] M.A. Zaks, A.S. Pikovsky, J. Kurths, J. Phys. A, Math. Gen. 32, 1523 (1999), doi: 10.1088/0305-4470/32/8/018
• [18] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York 1974; reprint Dover, New York 2006
• [19] M. Kac, Ann. Math. 47, 33 (1946), doi: 10.2307/1969033
• [20] G.H. Hardy, J.E. Littlewood, Acta Math. 37, 155 (1914), doi: 10.1007/BF02401833
• [21] N.J.A.S. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org
• [22] M. Baake, F. Gähler, U. Grimm, J. Math. Phys. 53, 032701 (2012), doi: 10.1063/1.3688337
• [23] J. Wolny, A. Wnęk, J.-L. Verger-Gaugry, J. Comput. Phys. 163, 313 (2000), doi: 10.1006/jcph.2000.6563
• [24] M. Baake, U. Grimm, Erg. Th. Dynam. Syst., arXiv:1205.1384

Identifiers

DOI

10.12693/APhysPolA.126.431