# A Comment on the Relation between Diffraction and Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Diffraction of Weighted Dirac Combs

## 3. Bernoullisation

**Theorem**

**1**

## 4. Close-Packed Dimers

**Theorem**

**2**

## 5. Ledrappier’s Model

**Theorem**

**3**

## 6. Meyer Sets with Entropy

**Theorem**

**4**

## 7. Visible Lattice Points

**Theorem**

**5**

## 8. Concluding Remarks

## References

- Cowley, J.M. Diffraction Physics, 3rd ed.; North-Holland: Amsterdam, The Netherlands, 1995. [Google Scholar]
- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.
**1984**, 53, 1951–1953. [Google Scholar] [CrossRef] - Cornfeld, I.P.; Fomin, S.V.; Sinai, Ya.G. Ergodic Theory; Springer: New York, NY, USA, 1982. [Google Scholar]
- Baake, M.; Lenz, D.; Richard, C. Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies. Lett. Math. Phys.
**2007**, 82, 61–77. [Google Scholar] [CrossRef] - Höffe, M.; Baake, M. Surprises in diffuse scattering. Z. Krist.
**2000**, 215, 441–444. [Google Scholar] [CrossRef] - Withers, R.L. Disorder, structured diffuse scattering and the transmission electron microscope. Z. Krist.
**2005**, 220, 1027–1034. [Google Scholar] [CrossRef] - Hof, A. On diffraction by aperiodic structures. Commun. Math. Phys.
**1995**, 169, 25–43. [Google Scholar] [CrossRef] - Hof, A. Diffraction by aperiodic structures at high temperatures. J. Phys. A: Math. Gen.
**1995**, 28, 57–62. [Google Scholar] [CrossRef] - Baake, M.; Moody, R.V. Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math.
**2004**, 573, 61–94. [Google Scholar] [CrossRef] - Baake, M.; Grimm, U. Kinematic diffraction from a mathematical viewpoint. Z. Krist.
**2011**, 226, 711–725. [Google Scholar] [CrossRef] - Berg, C.; Forst, G. Potential Theory on Locally Compact Abelian Groups; Springer: Berlin, Germany, 1975. [Google Scholar]
- Baake, M.; Grimm, U. Theory of Aperiodic Order: A Mathematical Invitation; Cambridge University Press: Cambridge, UK, in preparation.
- Baake, M.; Grimm, U. The singular continuous diffraction measure of the Thue-Morse chain. J. Phys. A.: Math. Theor.
**2008**, 41, 422001. [Google Scholar] [CrossRef] - Baake, M.; Gähler, F.; Grimm, U. Spectral and topological properties of a family of generalised Thue-Morse sequences. J. Math. Phys.
**2012**, 53, 032701. [Google Scholar] [CrossRef] - Baake, M.; Moody, R.V. Diffractive point sets with entropy. J. Phys. A: Math. Gen.
**1998**, 31, 9023–9039. [Google Scholar] [CrossRef] - Baake, M.; Birkner, M.; Moody, R.V. Diffraction of stochastic point sets: Explicitly computable examples. Commun. Math. Phys.
**2010**, 293, 611–660. [Google Scholar] [CrossRef] - Baake, M.; Grimm, U. Kinematic diffraction is insufficient to distinguish order from disorder. Phys. Rev. B
**2009**, 79, 020203(R), Erratum. Phys. Rev. B 2009, 80, 029903(E). [Google Scholar] [CrossRef] - Queffélec, M. Substitution Dynamical Systems—Spectral AnalysisLNM 1294, 2nd ed.; Springer: Berlin, Germany, 2010. [Google Scholar]
- Pytheas Fogg, N. Substitutions in Dynamics, Arithmetics and Combinatorics; LNM 1794; Springer: Berlin, Germany, 2002. [Google Scholar]
- Baake, M.; van Enter, A.C.D. Close-packed dimers on the line: Diffraction versus dynamical spectrum. J. Stat. Phys.
**2011**, 143, 88–101. [Google Scholar] [CrossRef] - van Enter, A.C.D.; Miȩkisz, J. How should one define a (weak) crystal? J. Stat. Phys.
**1992**, 66, 1147–1153. [Google Scholar] [CrossRef] - Schmidt, K. Dynamical Systems of Algebraic Origin; Birkhäuser: Basel, Switzerland, 1995. [Google Scholar]
- Ledrappier, F. Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A-B
**1987**, 287, A561–A563. [Google Scholar] - Baake, M.; Ward, T. Planar dynamical systems with pure Lebesgue diffraction spectrum. J. Stat. Phys.
**2010**, 140, 90–102. [Google Scholar] [CrossRef] - Meyer, Y. Algebraic Numbers and Harmonic Analysis; North Holland: Amsterdam, The Netherlands, 1972. [Google Scholar]
- Moody, R.V. Model sets: A survey. In From Quasicrystals to More Complex Systems; Axel, F., Dénoyer, F., Gazeau, J.-P., Eds.; EDP Sciences, Les Ulis, and Springer: Berlin, Germany, 2000; pp. 145–166. [Google Scholar]
- Lagarias, J.C. Meyer’s concept of quasicrystal and quasiregular sets. Commun. Math. Phys.
**1996**, 179, 365–376. [Google Scholar] [CrossRef] - Strungaru, N. Almost periodic measures and long-range order in Meyer sets. Discr. Comput. Geom.
**2005**, 33, 483–505. [Google Scholar] [CrossRef] - Baake, M. Diffraction of weighted lattice subsets. Can. Math. Bulletin
**2002**, 45, 483–498. [Google Scholar] [CrossRef] - Baake, M.; Moody, R.V.; Pleasants, P.A.B. Diffraction from visible lattice points and k-th power free integers. Discr. Math.
**2000**, 221, 3–42. [Google Scholar] [CrossRef] - Huck, C.; Pleasants, P.A.B. Entropy and diffraction of the k-free points in n-dimensional lattices. arXiv
**2011**. [Google Scholar] - Baake, M.; Lenz, D. Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra. Ergod. Theor. Dyn. Syst.
**2004**, 24, 1867–1893, math.DS/0302231. [Google Scholar] [CrossRef] - Gouéré, J.-B. Diffraction and Palm measure of point processes. C. R. Acad. Sci. Paris
**2003**, 342, 141–146. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Baake, M.; Grimm, U. A Comment on the Relation between Diffraction and Entropy. *Entropy* **2012**, *14*, 856-864.
https://doi.org/10.3390/e14050856

**AMA Style**

Baake M, Grimm U. A Comment on the Relation between Diffraction and Entropy. *Entropy*. 2012; 14(5):856-864.
https://doi.org/10.3390/e14050856

**Chicago/Turabian Style**

Baake, Michael, and Uwe Grimm. 2012. "A Comment on the Relation between Diffraction and Entropy" *Entropy* 14, no. 5: 856-864.
https://doi.org/10.3390/e14050856