Hostname: page-component-74656f9884-c52kr Total loading time: 0 Render date: 2026-11-09T12:31:13.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary processes and pure point diffraction

Published online by Cambridge University Press:  04 July 2016

DANIEL LENZ  and
ROBERT V. MOODY
DANIEL LENZ
Affiliation:
Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, Jena D-07743, Germany email daniel.lenz@uni-jena.de
ROBERT V. MOODY
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4, Canada email rmoody@uvic.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact abelian groups. These objects provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular, we can construct complete families of solutions to the inverse problem from any given positive pure point measure that is chosen to be the diffraction. In this case these processes can be classified by the dual of the group of relators based on the set of Bragg peaks, and this gives an abstract solution to the homometry problem for pure point diffraction.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2597 - 2642
Copyright
© Cambridge University Press, 2016 

References

Baake, M., Birkner, M. and Moody, R. V.. Diffraction of stochastic point sets: explicitly computable examples. Comm. Math. Phys. 293 (2010), 611–660.Google 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.Google Scholar
Baake, M., van Enter, A. C. D. and Lenz, D.. Dynamical versus diffraction spectrum for structures with finite local complexity. Ergod. Th. & Dynam. Sys. 35 (2015), 217–241.CrossRefGoogle Scholar
Baake, M. and Grimm, U.. Aperiodic Order (A Mathematical Invitation, 1) . Cambridge University Press, Cambridge, 2013.Google Scholar
Baake, M., Kösters, H. and Moody, R. V.. Diffraction theory of point processes: systems with clumping and repulsion. J. Stat. Phys. 159 (2015), 915–936.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
M. Baake and R. V. Moody (Eds). Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). American Mathematical Society, Providence, RI, 2000.Google Scholar
Bellissard, J., Hermann, D. and Zarrouati, M.. Hulls of aperiodic solids and gap labeling theorems. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). Eds. M. Baake and R. V. Moody. American Mathematical Society, Providence, RI, 2000, 207–259.CrossRefGoogle Scholar
Berg, C. and Forst, G.. Potential Theory on Locally Compact Abelian Groups. Springer, Berlin, 1975.Google Scholar
Daley, D. J. and Vere-Jones, D.. An Introduction to the Theory of Point Processes (Springer Series in Statistics) . Springer, New York, 1988.Google Scholar
Deng, X. and Moody, R. V.. Dworkin’s argument revisited: point processes, dynamics, diffraction, and correlations. J. Geom. Phys. 58 (2008), 506–541.Google Scholar
Dworkin, S.. Spectral theory and X-ray diffraction. J. Math. Phys. 34 (1993), 2965–2967.Google Scholar
van Enter, A. C. D. and Miȩkisz, J.. How should one define a (weak) crystal? J. Stat. Phys. 66 (1992), 1147–1153.Google Scholar
Gil de Lamadrid, J. and Argabright, L. N.. Almost Periodic Measures (Memoirs of the American Mathematical Society, 428) . American Mathematical Society, Providence, RI, 1990.Google Scholar
Gouéré, J.-B.. Diffraction et mesure de Palm des processus ponctuels. C. R. Math. Acad. Sci. Paris 336 (2003), 57–62.CrossRefGoogle Scholar
Gouéré, J.-B.. Quasicrystals and almost periodicity. Comm. Math. Phys. 255 (2005), 651–681.CrossRefGoogle Scholar
Grünbaum, F. A. and Moore, C. C.. The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets. Acta Crystallogr. A 51 (1995), 310–323.CrossRefGoogle Scholar
Hof, A.. On diffraction by aperiodic structures. Comm. Math. Phys. 169 (1995), 25–43.CrossRefGoogle Scholar
Karr, A. F.. Point Processes and their Statistical Inference. Marcel Dekker, New York, 1991.Google Scholar
Krengel, U.. Ergodic Theorems. de Gruyter, Berlin, 1985.Google Scholar
Lagarias, J.. Mathematical quasicrystals and the problem of diffraction. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13) . Eds. Baake, M. and Moody, R. V.. American Mathematical Society, Providence, RI, 2000, pp. 61–93.Google Scholar
Lagarias, J. and Pleasants, P. A. B.. Local complexity of Delone sets and crystallinity. Canad. Math. Bull. 45 (2002), 634–652.Google Scholar
Lagarias, J. and Pleasants, P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23 (2003), 831–867.Google Scholar
Lee, J.-Y., Moody, R. V. and Solomyak, B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3 (2002), 1003–1018.Google Scholar
Lenz, D.. Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Comm. Math. Phys. 287 (2009), 225–258.Google Scholar
Lenz, D. and Moody, R. V.. Extinctions and correlations for uniformly discrete point processes with pure point spectrum. Comm. Math. Phys. 289 (2009), 907–923.Google Scholar
Lenz, D. and Richard, C.. Pure point diffraction and cut and project schemes for measures: the smooth case. Math. Z. 256 (2007), 347–378.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.Google Scholar
Loomis, L. H.. An Introduction to Abstract Harmonic Analysis. Van Nostrand, Princeton, NJ, 1953.Google Scholar
Moody, R. V. and Strungaru, N.. Point sets and dynamical systems in the autocorrelation topology. Canad. Math. Bull. 47 (2004), 82–99.Google Scholar
Moody, R. V. and Strungaru, N.. Almost periodic measures and their Fourier transforms. Eds. Baake, M. and Grimm, U.. Preprint [to appear in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity].Google Scholar
Radin, C.. Miles of Tiles (Student Mathematical Library) . American Mathematical Society, Providence, RI, 1999.Google Scholar
Radin, C. and Wolff, M.. Space tilings and local isomorphism. Geom. Dedicata 42 (1992), 355–360.Google Scholar
Robinson, E. A.. The dynamical properties of Penrose tilings. Trans. Amer. Math. Soc. 348 (1996), 4447–4464.Google Scholar
Robinson, E. A.. Halmos–von Neumann theorem for model sets, and almost automorphic dynamical systems. Dynamics, Ergodic Theory, and Geometry (Mathematical Sciences Research Institute Publications, 54) . Cambridge University Press, Cambridge, 2007, pp. 243–272.Google Scholar
The Royal Institution, Crystallography Timeline.http://www.rigb.org/our-history/history-of-research/crystallography-timeline.Google Scholar
Schlottmann, M.. Generalized 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
Solomyak, B.. Spectrum of dynamical systems arising from Delone sets. Quasicrystals and Discrete Geometry (Fields Institute Monographs, 10) . Ed. Patera, J.. American Mathematical Society, Providence, RI, 1998, pp. 265–275.Google Scholar
Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695–738; Erratum: Ergod. Th. & Dynam. Sys. 19 (1999) 1685.Google Scholar
Strungaru, N.. Almost periodic measures and long-range order in Meyer sets. Discrete Comput. Geom. 33 (2005), 483–505.Google Scholar
Tempelman, A.. Ergodic Theorems for Group Actions. Kluwer, Dordrecht, 1992.CrossRefGoogle Scholar
Terauds, V.. The inverse problem of pure point diffraction—examples and open questions. J. Stat. Phys. 152 (2013), 954–968.Google Scholar
Terauds, V. and Baake, M.. Some comments in the inverse problem of pure point diffraction. Aperiodic Crystals. Eds. Schmidt, S., Withers, R. L. and Lifshitz, R.. Springer, Dordrecht, 2013, pp. 35–41.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, Berlin, 1982.Google Scholar
Withers, R. L.. ‘Disorder’: structured diffuse scattering and local crystal chemistry. Advances in Imaging and Electron Physics. Elsevier, San Diego, 2008, 152.Google Scholar