Hostname: page-component-89b8bd64d-j4x9h Total loading time: 0 Render date: 2026-05-09T08:30:18.057Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical versus diffraction spectrum for structures with finite local complexity

Published online by Cambridge University Press:  05 August 2014

MICHAEL BAAKE ,
DANIEL LENZ  and
AERNOUT VAN ENTER
MICHAEL BAAKE
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany email mbaake@math.uni-bielefeld.de
DANIEL LENZ
Affiliation:
Fakultät für Mathematik, Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany email daniel.lenz@uni-jena.de
AERNOUT VAN ENTER
Affiliation:
Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands email a.c.d.van.enter@rug.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2017 - 2043
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alexeyev, V. M.. Existence of a bounded function of the maximal spectral type. Ergod. Th. & Dynam. Sys. 2 (1982), 259–261 (originally published in Russian in 1958).CrossRefGoogle Scholar
Baake, M.. Diffraction of weighted lattice subsets. Can. Math. Bull. 45 (2002), 483–498.CrossRefGoogle Scholar
Baake, M., Birkner, M. and Moody, R V.. Diffraction of stochastic point sets: explicitly computable examples. Commun. Math. Phys. 293 (2009), 611–660.CrossRefGoogle Scholar
Baake, M. and van Enter, A. C. D.. Close-packed dimers on the line: diffraction versus dynamical spectrum. J. Stat. Phys. 143 (2011), 88–101.CrossRefGoogle Scholar
Baake, M., Gähler, F. and Grimm, U.. Spectral and topological properties of a family of generalised Thue–Morse sequences. J. Math. Phys. 53 (2012), 032701, 24 pp; arXiv:1201.1423.CrossRefGoogle Scholar
Baake, M., Gähler, F. and Grimm, U.. Examples of substitution systems and their factors. J. Integer Seq. 16 (2013), art. 13.2.14, 18 pp, arXiv:1211.5466.Google Scholar
Baake, M. and Grimm, U.. The singular continuous diffraction measure of the Thue–Morse chain. J. Phys. A 41 (2008), 422001, 6 pp; arXiv:0809.0580.CrossRefGoogle Scholar
Baake, M. and Grimm, U.. Kinematic diffraction from a mathematical viewpoint. Z. Krist. 226 (2011), 711–725.CrossRefGoogle Scholar
Baake, M. and Grimm, U.. Squirals and beyond: substitution tilings with singular continuous spectrum. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2012.191, published online 20 March 2013; arXiv:1205.1384.Google Scholar
Baake, M. and Grimm, U.. Aperiodic Order (A Mathematical Invitation, 1). Cambridge University Press, Cambridge, 2013.CrossRefGoogle Scholar
Baake, M. and Lenz, D.. Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Th. & Dynam. Sys. 24 (2004), 1867–1893.CrossRefGoogle Scholar
Baake, M. and Lenz, D.. Deformation of Delone dynamical systems and topological conjugacy. J. Fourier Anal. Appl. 11 (2005), 125–150.CrossRefGoogle Scholar
Baake, M., Lenz, D. and Moody, R. V.. Characterization of model sets by dynamical systems. Ergod. Th. & Dynam. Sys. 27 (2007), 341–382.CrossRefGoogle Scholar
Baake, M. and Moody, R. V.. Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. (Crelle) 573 (2004), 61–94.Google Scholar
Berg, C. and Forst, G.. Potential Theory on Locally Compact Abelian Groups. Springer, Berlin, 1975.CrossRefGoogle Scholar
Cornfeld, I. P., Fomin, S V. and Sinai, Ya. G.. Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
Cortez, M. I., Durand, F. and Petite, S.. Linearly repetitive Delone systems have a finite number of non-periodic Delone system factors. Proc. Amer. Math. Soc. 138 (2010), 1033–1046.CrossRefGoogle Scholar
Cowley, J. M.. Diffraction Physics, 3rd edn. North-Holland, Amsterdam, 1995.Google Scholar
Deng, X. and Moody, R. V.. Dworkins argument revisited: point processes, dynamics, diffraction, and correlations. J. Geom. Phys. 58 506–541.CrossRefGoogle Scholar
Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory of Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, 1976.CrossRefGoogle Scholar
Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20 (2000), 1061–1078.CrossRefGoogle Scholar
Dworkin, S.. Spectral theory and X-ray diffraction. J. Math. Phys. 34 (1993), 2965–2967.CrossRefGoogle Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory with a View towards Number Theory (Graduate Texts in Mathematics, 259). Springer, London, 2011.CrossRefGoogle Scholar
van Enter, A. C. D. and Miȩkisz, J.. How should one define a (weak) crystal? J. Stat. Phys. 66 (1992), 1147–1153.CrossRefGoogle Scholar
Fraczek, K. M.. On a function that realizes the maximal spectral type. Studia Math. 124 (1997), 1–7.CrossRefGoogle Scholar
Frank, N. P.. Multi-dimensional constant-length substitution sequences. Topol. Appl. 152 (2005), 44–69.CrossRefGoogle Scholar
Frettlöh, D. and Richard, C.. Dynamical properties of almost repetitive Delone sets. Discr. Cont. Dynam. Syst. 34 (2014), 531–556.Google Scholar
Halmos, P. R. and von Neumann, J.. Operator methods in classical mechanics. II. Ann. Math. 43 (1942), 332–350.CrossRefGoogle Scholar
Herning, J. L.. Spectrum and factors of substitution dynamical systems. PhD Thesis, George Washington University, Washington, DC 2013.Google Scholar
Hof, A.. On diffraction by aperiodic structures. Commun. Math. Phys. 169 (1995), 25–43.CrossRefGoogle Scholar
Koopman, B. O.. Hamiltonian systems and transformations in Hilbert space. Proc. Natl Acad. Sci. U.S.A. 17 (1931), 315–318.CrossRefGoogle ScholarPubMed
Lang, S.. Real and Functional Analysis, 3rd edn. Springer, New York, 1993.CrossRefGoogle Scholar
Lee, J. Y., Moody, R. V. and Solomyak, B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3 (2002), 1003–1018.CrossRefGoogle Scholar
Lenz, D.. Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287 (2009), 225–258.CrossRefGoogle Scholar
Lenz, D. and Moody, R. V.. Extinctions and correlations for uniformly discrete point processes with pure point dynamical spectra. Commun. Math. Phys. 289 (2009), 907–923.CrossRefGoogle Scholar
Lenz, D. and Moody, R. V.. Stationary processes with pure point diffraction. Preprint, 2011, arXiv: 1111.3617.Google Scholar
Lenz, D. and Strungaru, N.. Pure point spectrum for measure dynamical systems on locally compact Abelian groups. J. Math. Pures Appl. 92 (2009), 323–341.CrossRefGoogle Scholar
Lind, D. A. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Loomis, L. H.. An Introduction to Abstract Harmonic Analysis. Van Nostrand, Princeton, NJ, 1953, Reprint: Dover, New York, 2011.Google Scholar
Müller, P. and Richard, C.. Ergodic properties of randomly coloured point sets. Canad. J. Math. 65 (2013), 349–402.CrossRefGoogle Scholar
Nadkarni, M. G.. Spectral Theory of Dynamical Systems. Birkhäuser, Basel, 1998.Google Scholar
von Neumann, J.. Zur Operatorenmethode in der klassischen Mechanik. Ann. of Math. (2) 33 (1933), 587–642.CrossRefGoogle Scholar
Ornstein, D.. Factors of Bernoulli shifts are Bernoulli shifts. Adv. Math. 5 (1971), 349–364.CrossRefGoogle Scholar
Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116–136.CrossRefGoogle Scholar
Pedersen, G. K.. Analysis Now. Springer, New York, 1995, revised printing.Google Scholar
Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294), 2nd edn. Springer, Berlin, 2010.CrossRefGoogle Scholar
Robinson, E. A. Jr. Symbolic dynamics and tilings of ℝd. Proc. Sympos. Appl. Math. 60 (2004), 81–119.CrossRefGoogle Scholar
Rudin, W.. Fourier Analysis on Groups. Wiley, New York, 1962.Google Scholar
Rudolph, D. J.. Fundamentals of Measurable Dynamics. Clarendon Press, Oxford, 1990.Google Scholar
Schlottmann, M.. Generalised model sets and dynamical systems. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). Eds. Baake, M. and Moody, R. V.. American Mathematical Society, Providence, RI, 2000, pp. 143–159.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 2000, Reprint.Google Scholar
Withers, R. L.. Disorder, structured diffuse scattering and the transmission electron microscope. Z. Krist. 220 (2005), 1027–1034.CrossRefGoogle Scholar