Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T10:18:40.730Z Has data issue: false hasContentIssue false
  • English
  • Français

Delone dynamical systems and spectral convergence

Part of: Discrete geometry Locally compact groups and their algebras Selfadjoint operator algebras Classical measure theory

Published online by Cambridge University Press:  22 October 2018

SIEGFRIED BECKUS  and
FELIX POGORZELSKI
SIEGFRIED BECKUS
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel email beckus.siegf@technion.ac.il
FELIX POGORZELSKI
Affiliation:
Department of Mathematics, University of Leipzig, 04109 Leipzig, Germany email felix.pogorzelski@math.uni-leipzig.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

Keywords

topological dynamicsgroup actionsoperator algebrasHamiltonian dynamics

MSC classification

Primary: 28A33: Spaces of measures, convergence of measures 22D40: Ergodic theory on groups 52C23: Quasicrystals, aperiodic tilings
Secondary: 46L60: Applications of selfadjoint operator algebras to physics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1510 - 1544
Copyright
© Cambridge University Press, 2018

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.)

References

Avila, A. and Jitomirskaya, S.. The ten martini problem. Ann. of Math. (2) 170(1) (2009), 303–342.10.4007/annals.2009.170.303Google Scholar
Baake, M.. Diffraction of weighted lattice subsets. Canad. Math. Bull. 45(4) (2002), 483–498, dedicated to Robert V. Moody.10.4153/CMB-2002-050-2Google Scholar
Baake, M. and Grimm, U.. Aperiodic Order. Vol. 1 (Encyclopedia of Mathematics and its Applications, 149). Cambridge University Press, Cambridge, 2013, a mathematical invitation, with a foreword by Roger Penrose.10.1017/CBO9781139025256Google Scholar
Baake, M., Huck, C. and Strungaru, N.. On weak model sets of extremal density. Indag. Math. (N.S.) 28(1) (2017), 3–31.10.1016/j.indag.2016.11.002Google Scholar
Baake, M. and Lenz, D.. Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Th. & Dynam. Sys. 24(6) (2004), 1867–1893.10.1017/S0143385704000318Google Scholar
Baake, M. and Lenz, D.. Deformation of Delone dynamical systems and pure point diffraction. J. Fourier Anal. Appl. 11(2) (2005), 125–150.Google Scholar
Baake, M. and Moody, R. V.. Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573 (2004), 61–94.Google Scholar
Beckus, S.. Spectral approximation of aperiodic Schrödinger operators. PhD Thesis, Friedrich-Schiller-University Jena, Germany, 2016, https://arxiv.org/abs/1610.05894, p. 244.Google Scholar
Beckus, S. and Bellissard, J.. Continuity of the spectrum of a field of self-adjoint operators. Ann. Henri Poincaré 17(12) (2016), 3425–3442.10.1007/s00023-016-0496-3Google Scholar
Beckus, S., Bellissard, J. and Cornean, H.. Spectral stability of Schrödinger operators in the Hausdorff metric. Preprint, 2017.Google Scholar
Beckus, S., Bellissard, J. and de Nittis, G.. Spectral continuity for aperiodic quantum systems I. General theory. J. Funct. Anal. 275(11) (2018), 2917–2977.10.1016/j.jfa.2018.09.004Google Scholar
Beckus, S., Bellissard, J. and de Nittis, G.. Spectral continuity for aperiodic quantum systems II. Periodic approximations in 1Dm. Preprint, 2018, arXiv:1803.03099.Google Scholar
Beer, G.. On the Fell topology. Set-Valued Anal. 1(1) (1993), 69–80.10.1007/BF01039292Google Scholar
Bellissard, J.. K-theory of C ∗ -algebras in solid state physics. Statistical Mechanics and Field Theory: Mathematical Aspects (Groningen 1985) (Lecture Notes in Physics, 257). Springer, Berlin, 1986, pp. 99–156.10.1007/3-540-16777-3_74Google Scholar
Bellissard, J.. Spectral properties of Schrödinger’s operator with a Thue–Morse potential. Number Theory and Physics (Les Houches 1989) (Springer Proceedings in Physics, 47). Springer, Berlin, 1990, pp. 140–150.Google Scholar
Bellissard, J., Bovier, A. and Ghez, J. M.. Spectral properties of a tight binding Hamiltonian with period doubling potential. Comm. Math. Phys. 135(2) (1991), 379–399.10.1007/BF02098048Google Scholar
Bellissard, J., Herrmann, D. J. L. and Zarrouati, M.. Hulls of aperiodic solids and gap labeling theorems. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). Providence, RI, 2000, pp. 207–258.Google Scholar
Bellissard, J., Iochum, B., Scoppola, E. and Testard, D.. Spectral properties of one-dimensional quasi-crystals. Comm. Math. Phys. 125(3) (1989), 527–543.10.1007/BF01218415Google Scholar
Bellissard, J., Iochum, B. and Testard, D.. Continuity properties of the electronic spectrum of 1D quasicrystals. Comm. Math. Phys. 141(2) (1991), 353–380.10.1007/BF02101510Google Scholar
Björklund, M. and Hartnick, T.. Approximate lattices. Duke Math. J. (2018), 2016, to appear, arXiv:1612.09246.10.1215/00127094-2018-0028Google Scholar
Björklund, M., Hartnick, T. and Pogorzelski, F.. Aperiodic order and spherical diffraction II: The shadow transform and the diffraction formula. Preprint, 2017, arXiv:1704.00302.Google Scholar
Björklund, M., Hartnick, T. and Pogorzelski, F.. Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets. Proc. Lond. Math. Soc. (3) 116(4) (2018), 957–996.10.1112/plms.12091Google Scholar
Bourbaki, N.. Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et Représentations (Actualités Scientifiques et Industrielles, 1306). Hermann, Paris, 1963.Google Scholar
Bridson, M. R., de la Harpe, P. and Kleptsyn, V. The Chabauty space of closed subgroups of the three-dimensional Heisenberg group. Pacific J. Math. 240(1) (2009), 1–48.10.2140/pjm.2009.240.1Google Scholar
Chabauty, C.. Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78 (1950), 143–151.10.24033/bsmf.1412Google Scholar
Connes, A.. Sur la théorie non commutative de l’intégration. Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) (Lecture Notes in Mathematics, 725). Springer, Berlin, 1979, pp. 19–143.10.1007/BFb0062614Google Scholar
Damanik, D.. Singular continuous spectrum for a class of substitution Hamiltonians. Lett. Math. Phys. 46(4) (1998), 303–311.10.1023/A:1007510721504Google Scholar
Damanik, D. and Gorodetski, A.. Sums of regular cantor sets of large dimension and the square Fibonacci Hamiltonian. Preprint, 2016, arXiv:1601.01639.Google Scholar
Damanik, D., Gorodetski, A. and Solomyak, B.. Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian. Duke Math. J. 164(8) (2015), 1603–1640.10.1215/00127094-3119739Google Scholar
Damanik, D., Gorodetski, A. and Yessen, W.. The Fibonacci Hamiltonian. Invent. Math. 206(3) (2016), 629–692.Google Scholar
Damanik, D. and Lenz, D.. Uniform spectral properties of one-dimensional quasicrystals. I. Absence of eigenvalues. Comm. Math. Phys. 207(3) (1999), 687–696.Google Scholar
Damanik, D. and Lenz, D.. Uniform spectral properties of one-dimensional quasicrystals. II. The Lyapunov exponent. Lett. Math. Phys. 50(4) (1999), 245–257.10.1023/A:1007614218486Google Scholar
Deaconu, V.. Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347(5) (1995), 1779–1786.10.1090/S0002-9947-1995-1233967-5Google Scholar
Deitmar, A. and Echterhoff, S.. Principles of Harmonic Analysis (Universitext). Springer, New York, 2009.Google Scholar
Dixmier, J.. von Neumann Algebras (North–Holland Mathematical Library, 27). North–Holland, Amsterdam, 1981, with a preface by E. C. Lance, translated from the second French edition by F. Jellett.Google Scholar
Dworkin, S.. Spectral theory and x-ray diffraction. J. Math. Phys. 34(7) (1993), 2965–2967.Google Scholar
Edagawa, K., Suzuki, K. and Takeuchi, S.. High resolution transmission electron microscopy observation of thermally fluctuating phasons in decagonal al-cu-co. Phys. Rev. Lett. 85 (2000), 1674–1677.10.1103/PhysRevLett.85.1674Google Scholar
Elek, G.. L 2 -spectral invariants and convergent sequences of finite graphs. J. Funct. Anal. 254(10) (2008), 2667–2689.10.1016/j.jfa.2008.01.010Google Scholar
Fell, J. M. G.. A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc. 13 (1962), 472–476.10.1090/S0002-9939-1962-0139135-6Google Scholar
Gelander, T.. A lecture on invariant random subgroups. New Directions in Locally Compact Groups (London Mathematical Society Lecture Note Series). Eds. Caprace, P.-E. and Monod, N.. Cambridge University Press, Cambridge, 2018, pp. 186–204.Google Scholar
Gouéré, J.-B.. Quasicrystals and almost periodicity. Comm. Math. Phys. 255(3) (2005), 655–681.10.1007/s00220-004-1271-8Google Scholar
Hausdorff, F.. Grundzüge der Mengenlehre. Reprinted 1949 by Chelsea Publishing Company, New York, 1914.Google Scholar
Hof, A.. On diffraction by aperiodic structures. Comm. Math. Phys. 169(1) (1995), 25–43.Google Scholar
Huck, C. and Richard, C.. On pattern entropy of weak model sets. Discrete Comput. Geom. 54(3) (2015), 741–757.Google Scholar
Kalugin, P. A.. A mechanism for diffusion in quasicrystals. Beyond Quasicrystals (Les Houches 1994). Springer, Berlin, 1995, pp. 191–201.Google Scholar
Kalugin, P. A. and Katz, A.. A mechanism for self-diffusion in quasi-crystals. Europhys. Lett. 21(9) (1993), 921.Google Scholar
Kellendonk, J.. Pattern-equivariant functions and cohomology. J. Phys. A 36(21) (2003), 5765–5772.Google Scholar
Kellendonk, J.. Pattern equivariant functions, deformations and equivalence of tiling spaces. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1153–1176.Google Scholar
Kellendonk, J. and Lenz, D.. Equicontinuous Delone dynamical systems. Canad. J. Math. 65(1) (2013), 149–170.Google Scholar
Kellendonk, J. and Putnam, I. F.. Tilings, C ∗ -algebras, and K-theory. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). American Mathematical Society, Providence, RI, 2000, pp. 177–206.Google Scholar
Keller, G. and Richard, C.. Dynamics on the graph of the torus parametrization. Ergod. Th. & Dynam. Sys. 38(3) (2018), 1048–1085.Google Scholar
Kramer, P. and Neri, R.. On periodic and nonperiodic space fillings of Em obtained by projection. Acta Cryst. Sect. A 40(5) (1984), 580–587.Google Scholar
Lagarias, J. C. and Pleasants, P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23(3) (2003), 831–867.Google Scholar
Lee, J.-Y., Moody, R. V. and Solomyak, B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5) (2002), 1003–1018.Google Scholar
Lenz, D.. Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals. Comm. Math. Phys. 227(1) (2002), 119–130.Google Scholar
Lenz, D. and Moody, R. V.. Extinctions and correlations for uniformly discrete point processes with pure point dynamical spectra. Comm. Math. Phys. 289(3) (2009), 907–923.Google Scholar
Lenz, D., Müller, P. and Veselić, I.. Uniform existence of the integrated density of states for models on ℤd. Positivity 12(4) (2008), 571–589.Google Scholar
Lenz, D., Peyerimhoff, N. and Veselić, I.. Groupoids, von Neumann algebras and the integrated density of states. Math. Phys. Anal. Geom. 10(1) (2007), 1–41.Google Scholar
Lenz, D. and Richard, C.. Pure point diffraction and cut and project schemes for measures: the smooth case. Math. Z. 256(2) (2007), 347–378.Google Scholar
Lenz, D., Schwarzenberger, F. and Veselić, I.. A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states. Geom. Dedicata 150 (2011), 1–34;  Erratum in doi:10.1007/s10711-011-9657-1.Google Scholar
Lenz, D. and Stollmann, P.. Algebras of random operators associated to Delone dynamical systems. Math. Phys. Anal. Geom. 6(3) (2003), 269–290.Google Scholar
Lenz, D. and Stollmann, P.. Delone dynamical systems and associated random operators. Operator Algebras and Mathematical Physics (Constanţa 2001). Theta, Bucharest, 2003, pp. 267–285.Google Scholar
Lenz, D. and Stollmann, P.. An ergodic theorem for Delone dynamical systems and existence of the integrated density of states. J. Anal. Math. 97 (2005), 1–24.Google Scholar
Lenz, D. and Stollmann, P.. Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians. Duke Math. J. 131(2) (2006), 203–217.Google Scholar
Lenz, D. and Strungaru, N.. Pure point spectrum for measure dynamical systems on locally compact abelian groups. J. Math. Pures Appl. (9) 92(4) (2009), 323–341.Google Scholar
Lenz, D. and Veselić, I.. Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence. Math. Z. 263(4) (2009), 813–835.Google Scholar
Levine, D. and Steinhardt, Paul J.. Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53 (1984), 2477–2480.Google Scholar
Mahler, K.. On lattice points in n-dimensional star bodies. I. Existence theorems. Proc. R. Soc. Lond. A 187 (1946), 151–187.Google Scholar
Meyer, Y.. À propos de la formule de Poisson. Séminaire d’Analyse Harmonique (année 1968/1969). Exp. No. 2. Faculté des Sciences d’Orsay – Université Paris-Sud, Paris, 1969 (in French).Google Scholar
Meyer, Y.. Nombres de Pisot, Nombres de Salem et Analyse Harmonique (Lecture Notes in Mathematics, 117). Springer, Berlin, 1970, Cours Peccot donné au Collège de France en avril-mai 1969.Google Scholar
Meyer, Y.. Algebraic Numbers and Harmonic Analysis. Vol. 2 (North-Holland Mathematical Library). North-Holland, Amsterdam, 1972.Google Scholar
Moody, R. V.. Meyer sets and the finite generation of quasicrystals. Symmetries in Science, VIII (Bregenz 1994). Plenum, New York, 1995, pp. 379–394.Google Scholar
Moody, R. V.. Meyer sets and their duals. The Mathematics of Long-Range Aperiodic Order (Waterloo, ON 1995) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489). Kluwer Academic, Dordrecht, 1997, pp. 403–441.Google Scholar
Moody, R. V.. Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130.Google Scholar
Mumford, D.. A remark on Mahler’s compactness theorem. Proc. Amer. Math. Soc. 28 (1971), 289–294.Google Scholar
Oler, N.. Spaces of closed subgroups of a connected Lie group. Glasgow Math. J. 14 (1973), 77–79.Google Scholar
Ostlund, S. and Kim, S.-H.. Renormalization of quasiperiodic mappings. Phys. Scripta T9 (1985), 193–198.Google Scholar
Pastur, L. A.. Selfaverageability of the number of states of the Schrödinger equation with a random potential. Mat. Fiz. i Funkcional. Anal. Vyp. 2 (1971), 111–116, 238.Google Scholar
Pogorzelski, F.. Convergence theorems for graph sequences. Internat. J. Algebra Comput. 24(8) (2014), 1233–1251.Google Scholar
Pogorzelski, F. and Schwarzenberger, F.. A Banach space-valued ergodic theorem for amenable groups and applications. J. Anal. Math. 130 (2016), 19–69.Google Scholar
Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.Google Scholar
Raghunathan, M. S.. Discrete Subgroups of Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68). Springer, New York, 1972.Google Scholar
Renault, J.. A Groupoid Approach to C ∗ -Algebras (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.Google Scholar
Richard, C. and Strungaru, N.. A short guide to pure point diffraction in cut-and-project sets. J. Phys. A 50(15) (2017),154003, 25.Google Scholar
Richard, C. and Strungaru, N.. Pure point diffraction and poisson summation. Ann. Henri Poincaré 18(12) (2017), 3903–3931.Google Scholar
Roe, J.. An index theorem on open manifolds. I, II. J. Differential Geom. 27(1) (1988), 87–113, 115–136.Google Scholar
Roe, J.. Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Amer. Math. Soc. 104(497) (1993), x+90.Google Scholar
Schlottmann, M.. Generalized model sets and dynamical systems. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). American Mathematical Society, Providence, RI, 2000, pp. 143–159.Google Scholar
Schumacher, C. and Schwarzenberger, F.. Approximation of the integrated density of states on sofic groups. Ann. Henri Poincaré 16(4) (2015), 1067–1101.Google Scholar
Shechtman, D., Blech, I., Gratias, D. and Cahn, J. W.. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), 1951–1953.Google Scholar
Shubin, M. A.. Spectral theory and the index of elliptic operators with almost-periodic coefficients. Russ. Math. Surveys 34(2) (1979), 109–157.Google Scholar
Strungaru, N.. On weighted Dirac combs supported inside model sets. J. Phys. A 47(33) (2014), 335202, 19.Google Scholar
Sütő, A.. The spectrum of a quasiperiodic Schrödinger operator. Comm. Math. Phys. 111(3) (1987), 409–415.Google Scholar
Vietoris, L.. Bereiche zweiter Ordnung. Monatsh. Math. Phys. 32(1) (1922), 258–280.Google Scholar
von Querenburg, B.. Mengentheoretische Topologie, 3rd edn. Springer, Berlin, 2001.Google Scholar