On weakly almost periodic measures

Lenz, Daniel; Strungaru, Nicolae

doi:10.1090/tran/7422

On weakly almost periodic measures
HTML articles powered by AMS MathViewer

by Daniel Lenz and Nicolae Strungaru PDF

Trans. Amer. Math. Soc. 371 (2019), 6843-6881 Request permission

Abstract:

We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point, respectively, continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier–Bohr coefficients of weakly almost periodic measures.

References

E. Akin and E. Glasner, WAP Systems and Labeled Subshifts, to appear in Memoirs of the Amer. Math. Soc. arXiv:1410.4753.
Loren Argabright and Jesûs Gil de Lamadrid, Fourier analysis of unbounded measures on locally compact abelian groups, Memoirs of the American Mathematical Society, No. 145, American Mathematical Society, Providence, R.I., 1974. MR 0621876
Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
J.-B. Aujogue, Pure Point/Continuous Decomposition of Translation-Bounded Measures and Diffraction, Ergodic Theory and Dynamical Systems, to appear. arXiv:1510.06381.
Jean-Baptiste Aujogue, Marcy Barge, Johannes Kellendonk, and Daniel Lenz, Equicontinuous factors, proximality and Ellis semigroup for Delone sets, Mathematics of aperiodic order, Progr. Math., vol. 309, Birkhäuser/Springer, Basel, 2015, pp. 137–194. MR 3381481, DOI 10.1007/978-3-0348-0903-0_{5}
Michael Baake and Franz Gähler, Pair correlations of aperiodic inflation rules via renormalisation: some interesting examples, Topology Appl. 205 (2016), 4–27. MR 3493304, DOI 10.1016/j.topol.2016.01.017
Michael Baake and Uwe Grimm, Aperiodic order. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 149, Cambridge University Press, Cambridge, 2013. A mathematical invitation; With a foreword by Roger Penrose. MR 3136260, DOI 10.1017/CBO9781139025256
Michael Baake and Christian Huck, Ergodic properties of visible lattice points, Proc. Steklov Inst. Math. 288 (2015), no. 1, 165–188. Reprint of Tr. Mat. Inst. Steklova 288 (2015), 184–208. MR 3485709, DOI 10.1134/S0081543815010137
Michael Baake, Christian Huck, and Nicolae Strungaru, On weak model sets of extremal density, Indag. Math. (N.S.) 28 (2017), no. 1, 3–31. MR 3597030, DOI 10.1016/j.indag.2016.11.002
Michael Baake and Daniel Lenz, Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems 24 (2004), no. 6, 1867–1893. MR 2106769, DOI 10.1017/S0143385704000318
Michael Baake and Daniel Lenz, Spectral notions of aperiodic order, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 2, 161–190. MR 3600642, DOI 10.3934/dcdss.2017009
Michael Baake, Daniel Lenz, and Robert V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 341–382. MR 2308136, DOI 10.1017/S0143385706000800
Michael Baake and Robert V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. 573 (2004), 61–94. MR 2084582, DOI 10.1515/crll.2004.064
S. Beckus, D. Lenz, F. Pogorzelski, and M. Schmidt, Diffraction theory for processes of tempered distributions, in preparation.
Christian Berg and Gunnar Forst, Potential theory on locally compact abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87, Springer-Verlag, New York-Heidelberg, 1975. MR 0481057
C. Corduneanu, Almost Periodic Functions, 2nd ed., Chelsea Publ. Co, New York, 1989.
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
Xinghua Deng and Robert V. Moody, Dworkin’s argument revisited: point processes, dynamics, diffraction, and correlations, J. Geom. Phys. 58 (2008), no. 4, 506–541. MR 2406412, DOI 10.1016/j.geomphys.2007.12.006
Tomasz Downarowicz and Eli Glasner, Isomorphic extensions and applications, Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 321–338. MR 3586277, DOI 10.12775/TMNA.2016.050
Steven Dworkin, Spectral theory and x-ray diffraction, J. Math. Phys. 34 (1993), no. 7, 2965–2967. MR 1224190, DOI 10.1063/1.530108
W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240. MR 36455, DOI 10.1090/S0002-9947-1949-0036455-9
W. F. Eberlein, A note on Fourier-Stieltjes transforms, Proc. Amer. Math. Soc. 6 (1955), 310–312. MR 68030, DOI 10.1090/S0002-9939-1955-0068030-2
W. F. Eberlein, The point spectrum of weakly almost periodic functions, Michigan Math. J. 3 (1955/56), 137–139. MR 82627
R. Ellis and M. Nerurkar, Weakly almost periodic flows, Trans. Amer. Math. Soc. 313 (1989), no. 1, 103–119. MR 930084, DOI 10.1090/S0002-9947-1989-0930084-3
S. Favorov, Bohr and Besicovitch almost periodic discrete sets and quasicrystals, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1761–1767. MR 2869161, DOI 10.1090/S0002-9939-2011-11046-3
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753, DOI 10.1090/surv/101
Jean-Baptiste Gouéré, Quasicrystals and almost periodicity, Comm. Math. Phys. 255 (2005), no. 3, 655–681. MR 2135448, DOI 10.1007/s00220-004-1271-8
U. Grimm and M. Baake, Homometric Point Sets and Inverse Problems, Z. Krist. 223 , 777-781, 2008. arXiv:0808.0094.
A. Hof, On diffraction by aperiodic structures, Comm. Math. Phys. 169 (1995), no. 1, 25–43. MR 1328260
A. Hof, Diffraction by aperiodic structures, The mathematics of long-range aperiodic order (Waterloo, ON, 1995) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 239–268. MR 1460026
S. Kasjan, G. Keller, and M. Lemanczyk, Dynamics of B-free sets: A view through the window, to appear in International Mathematics Research Notices, arXiv:1702.02375.
Johannes Kellendonk and Daniel Lenz, Equicontinuous Delone dynamical systems, Canad. J. Math. 65 (2013), no. 1, 149–170. MR 3004461, DOI 10.4153/CJM-2011-090-3
Johannes Kellendonk, Daniel Lenz, and Jean Savinien (eds.), Mathematics of aperiodic order, Progress in Mathematics, vol. 309, Birkhäuser/Springer, Basel, 2015. MR 3380566, DOI 10.1007/978-3-0348-0903-0
G. Keller and C. Richard, Dynamics on the graph of the torus parametrisation, to appear in Ergod. Th. & Dynam. Sys. arXiv:1511.06137.
J. C. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom. 21 (1999), no. 2, 161–191. MR 1668082, DOI 10.1007/PL00009413
J. C. Lagarias, Geometric models for quasicrystals. II. Local rules under isometries, Discrete Comput. Geom. 21 (1999), no. 3, 345–372. MR 1672976, DOI 10.1007/PL00009426
Jeffrey C. Lagarias, Mathematical quasicrystals and the problem of diffraction, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 61–93. MR 1798989
Jeffrey C. Lagarias and Peter A. B. Pleasants, Local complexity of Delone sets and crystallinity, Canad. Math. Bull. 45 (2002), no. 4, 634–652. Dedicated to Robert V. Moody. MR 1941231, DOI 10.4153/CMB-2002-058-0
Jeffrey C. Lagarias and Peter A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 831–867. MR 1992666, DOI 10.1017/S0143385702001566
Jesús Gil de Lamadrid and Loren N. Argabright, Almost periodic measures, Mem. Amer. Math. Soc. 85 (1990), no. 428, vi+219. MR 979431, DOI 10.1090/memo/0428
J.-Y. Lee, R. V. Moody, and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré 3 (2002), no. 5, 1003–1018. MR 1937612, DOI 10.1007/s00023-002-8646-1
D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces, arXiv:1608.05636.
Daniel Lenz and Robert V. Moody, Extinctions and correlations for uniformly discrete point processes with pure point dynamical spectra, Comm. Math. Phys. 289 (2009), no. 3, 907–923. MR 2511655, DOI 10.1007/s00220-009-0818-0
Daniel Lenz and Robert V. Moody, Stationary processes and pure point diffraction, Ergodic Theory Dynam. Systems 37 (2017), no. 8, 2597–2642. MR 3719270, DOI 10.1017/etds.2016.12
Daniel Lenz and Christoph Richard, Pure point diffraction and cut and project schemes for measures: the smooth case, Math. Z. 256 (2007), no. 2, 347–378. MR 2289878, DOI 10.1007/s00209-006-0077-0
Daniel Lenz and Nicolae Strungaru, Pure point spectrum for measure dynamical systems on locally compact abelian groups, J. Math. Pures Appl. (9) 92 (2009), no. 4, 323–341 (English, with English and French summaries). MR 2569181, DOI 10.1016/j.matpur.2009.05.013
D. Lenz and N. Strungaru, On uniquely ergodicity of systems of measures, in preparation.
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
Yves Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, Afr. Diaspora J. Math. 13 (2012), no. 1, 1–45. MR 2876415
Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403–441. MR 1460032
R. V. Moody, Model sets: A Survey, in: From Quasicrystals to More Complex Systems, (eds. F. Axel, F. Dénoyer and J. P. Gazeau), EDP Sciences, Les Ulis, and Springer, Berlin , pp. 145–166, 2000. math.MG/0002020.
Robert V. Moody and Nicolae Strungaru, Point sets and dynamical systems in the autocorrelation topology, Canad. Math. Bull. 47 (2004), no. 1, 82–99. MR 2032271, DOI 10.4153/CMB-2004-010-8
Robert V. Moody and Nicolae Strungaru, Almost periodic measures and their Fourier transforms, Aperiodic order. Vol. 2, Encyclopedia Math. Appl., vol. 166, Cambridge Univ. Press, Cambridge, 2017, pp. 173–270. MR 3791850
G. K. Pedersen, Analysis Now, Springer, New York, 1989; Revised printing (1995).
Charles Radin, Miles of tiles, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 237–258. MR 1411221, DOI 10.1017/CBO9780511662812.008
Hans Reiter and Jan D. Stegeman, Classical harmonic analysis and locally compact groups, 2nd ed., London Mathematical Society Monographs. New Series, vol. 22, The Clarendon Press, Oxford University Press, New York, 2000. MR 1802924
Christoph Richard, Dense Dirac combs in Euclidean space with pure point diffraction, J. Math. Phys. 44 (2003), no. 10, 4436–4449. MR 2008926, DOI 10.1063/1.1609032
Christoph Richard and Nicolae Strungaru, A short guide to pure point diffraction in cut-and-project sets, J. Phys. A 50 (2017), no. 15, 154003, 25. MR 3628773, DOI 10.1088/1751-8121/aa5d44
Martin Schlottmann, Generalized model sets and dynamical systems, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 143–159. MR 1798991
Boris Solomyak, Spectrum of dynamical systems arising from Delone sets, Quasicrystals and discrete geometry (Toronto, ON, 1995) Fields Inst. Monogr., vol. 10, Amer. Math. Soc., Providence, RI, 1998, pp. 265–275. MR 1636783, DOI 10.1090/fim/010/10
Nicolae Strungaru, Almost periodic measures and long-range order in Meyer sets, Discrete Comput. Geom. 33 (2005), no. 3, 483–505. MR 2121992, DOI 10.1007/s00454-004-1156-9
Nicolae Strungaru, Almost periodic measures and Bragg diffraction, J. Phys. A 46 (2013), no. 12, 125205, 11. MR 3035351, DOI 10.1088/1751-8113/46/12/125205
Nicolae Strungaru, On weighted Dirac combs supported inside model sets, J. Phys. A 47 (2014), no. 33, 335202, 19. MR 3247845, DOI 10.1088/1751-8113/47/33/335202
Nicolae Strungaru, Almost periodic pure point measures, Aperiodic order. Vol. 2, Encyclopedia Math. Appl., vol. 166, Cambridge Univ. Press, Cambridge, 2017, pp. 271–342. MR 3791851
Nicolae Strungaru and Venta Terauds, Diffraction theory and almost periodic distributions, J. Stat. Phys. 164 (2016), no. 5, 1183–1216. MR 3534490, DOI 10.1007/s10955-016-1579-8
Venta Terauds, The inverse problem for pure point diffraction—examples and open questions, J. Stat. Phys. 152 (2013), no. 5, 954–968. MR 3101489, DOI 10.1007/s10955-013-0790-0
V. Terauds and M. Baake, Some comments on the inverse problem of pure point diffraction, in Aperiodic Crystals, (eds. S. Schmid, R.L. Withers and R. Lifshitz), Springer, Dordrecht, pp. 35–41, 2013. arXiv:1210.3460.
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108