Diffraction of return time measures. (English) Zbl 1409.37011
The diffraction spectrum of a family of measures associated to ergodic transformations of the unit interval and weighting functions on the interval are studied. The measures are constructed as weighted Dirac combs along orbits of a reference point. In contrast to regular model sets arising from cut and project schemes, under quite mild conditions on the underlying map and on the weighting function, explicit calculations of the diffraction spectrum show the occurrence of an absolutely continuous contribution for almost every reference point. For rotations it is shown that the diffraction spectrum is a pure point for almost every reference point (and is so for all reference points for an ergodic rotation and a reasonable weighting function). Some new results on the limiting properties of diffraction spectra associated to convergent sequences of rotation numbers are found. Examples are constructed to explain the relationship between these results and those arising in the recent work of S. Beckus and F. Pogorzelski [“Delone dynamical systems and spectral convergence”, Ergodic Theory Dynam. Sys., to appear; doi:10.1017/etds.2018.116].
Reviewer: Thomas B. Ward (Leeds)
MSC:
| 37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |
| 43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 37E05 | Dynamical systems involving maps of the interval |
| 37A25 | Ergodicity, mixing, rates of mixing |
Keywords:
aperiodic order; autocorrelation; diffraction; transformations of the unit interval; rigid rotationsReferences:
| [1] | Argabright, L., Gil de Lamadrid, J.: Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups. Memoirs of the American Mathematical Society, No. 145. American Mathematical Society, Providence (1974) · Zbl 0294.43002 |
| [2] | Baake, M., Grimm, U.: Kinematic diffraction from a mathematical viewpoint. Z. Kristallogr. 226(9), 711-725 (2011) |
| [3] | Baake, M., Grimm, U.: Aperiodic Order. Cambridge University Press, CPI Group Ltd., London (2013) · Zbl 1295.37001 |
| [4] | Baake, M., Höffe, M.: Diffraction of random tilings: some rigorous results. J. Stat. Phys. 99(1-2), 216-261 (2000) · Zbl 1101.82345 |
| [5] | Baake, M., Lenz, D.: Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Theory Dyn. Syst. 24(6), 1867-1893 (2004) · Zbl 1127.37004 |
| [6] | Baake, M., Moody, R.V.: Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573, 61-94 (2004) · Zbl 1188.43008 |
| [7] | Beckus, S., Pogorzelski, F.: Delone dynamical systems and spectral convergence. Ergod. Theory Dyn. Syst. https://doi.org/10.1017/etds.2018.116 · Zbl 1443.37007 |
| [8] | Berend, D., Radin, C.: Are there chaotic tilings? Commun. Math. Phys. 152(2), 215-219 (1993) · Zbl 0790.28011 |
| [9] | Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathamatik und ihrer Grenzbebiete, Band. 87. Springer, Berlin (1975) · Zbl 0308.31001 |
| [10] | Boshernitzan, M.: A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52(3), 723-752 (1985) · Zbl 0602.28009 |
| [11] | Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer, London (2011) · Zbl 1206.37001 |
| [12] | Gil de Lamadrid, J., Argabright, L.: Almost periodic measures. Mem. Am. Math. Soc. 85(428), vi+219 (1990) · Zbl 0719.43006 |
| [13] | Gouéré, J.-B.: Diffraction and Palm measure of point processes (2002) |
| [14] | Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169(1), 25-43 (1995) · Zbl 0821.60099 |
| [15] | Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180(1), 119-140 (1982) · Zbl 0485.28016 |
| [16] | Ishimasa, T., Nissen, H.-U., Fukano, Y.: New ordered state between crystalline and amorphous in Ni-Cr particles. Phys. Rev. Lett. 55, 511-513 (1985) |
| [17] | Kellendonk, J., Lenz, D., Savinien, J. (eds.): Mathematics of Aperiodic Order. Progress in Mathematics, vol. 309. Birkhäuser/Springer, Basel (2015) · Zbl 1338.37005 |
| [18] | Keller, G.: On the rate of convergence to equilibrium in one-dimensional systems. Commun. Math. Phys. 96(2), 181-193 (1984) · Zbl 0576.58016 |
| [19] | Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. A Wiley-Interscience publication. Wiley, New York (1974) · Zbl 0281.10001 |
| [20] | Lenz, D.: An autocorrelation and discrete spectrum for dynamical systems on metric spaces. arXiv:1608.05636 (2016) |
| [21] | Lenz, D., Strungaru, N.: Pure point spectrum for measure dynamical systems on locally compact Abelian groups. J. Math. Pures Appl. (9) 92(4), 323-341 (2009) · Zbl 1178.37008 |
| [22] | Lenz, D., Strungaru, N.: On weakly almost periodic measures. Trans. Am. Math. Soc. https://doi.org/10.1090/tran/7422 · Zbl 1341.82081 |
| [23] | Lindenstrauss, E.: Pointwise theorems for amenable groups. Invent. Math. 146(2), 259-295 (2001) · Zbl 1038.37004 |
| [24] | Moody, R.V. (ed.): The Mathematics of Long-Range Aperiodic Order. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 489. Kluwer Academic Publishers Group, Dordrecht (1997) · Zbl 0861.00015 |
| [25] | Moody, R.V.: Mathematical quasicrystals: a tale of two topologies. In: XIVth International Congress on Mathematical Physics, pp. 68-77. World Scientific Publishing, Hackensack (2005) · Zbl 1106.52004 |
| [26] | Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. d’Anal. Math. 53(1), 139-186 (1989) · Zbl 0745.52013 |
| [27] | Müller, P., Richard, C.: Ergodic properties of randomly coloured point sets. Can. J. Math. 65(2), 349-402 (2013) · Zbl 1351.37030 |
| [28] | Richard, C.: Dense Dirac combs in Euclidean space with pure point diffraction. J. Math. Phys. 44(10), 4436-4449 (2003) · Zbl 1062.52022 |
| [29] | Richard, C., Strungaru, N.: Pure point diffraction and Poisson summation. Ann. Henri Poincaré 18(12), 3903-3931 (2017) · Zbl 1388.78003 |
| [30] | Richard, C., Strungaru, N.: A short guide to pure point diffraction in cut-and-project sets. J. Phys. A 50(15), 154003, 25 (2017) · Zbl 1366.82058 |
| [31] | Rudin, W.: Fourier Analysis on Groups. A Wiley-Interscience publication. Wiley, New York (1990) · Zbl 0698.43001 |
| [32] | Schlottmann, M.: Generalized model sets and dynamical systems. In: Directions in Mathematical Quasicrystals. CRM Monograph Series, vol. 13, pp. 143-159. American Mathematical Society, Providence (2000) · Zbl 0984.37005 |
| [33] | Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995) · Zbl 0828.52007 |
| [34] | Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951-1953 (1984) |
| [35] | Solomyak, B.: Dynamics of self-similar tilings. Ergod. Theory Dyn. Syst. 17(3), 695-738 (1997) · Zbl 0884.58062 |
| [36] | Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York (1982) · Zbl 0475.28009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
