Scaling of the Thue-Morse diffraction measure

@article{Baake2013ScalingOT,
  title={Scaling of the Thue-Morse diffraction measure},
  author={Michael Baake and Uwe Grimm and Johan Nilsson},
  journal={Acta Physica Polonica A},
  year={2013},
  volume={126},
  pages={431-434}
}
We revisit the well-known and much studied Riesz product representation of the Thue-Morse diffraction measure, which is also the maximal spectral measure for the corresponding dynamical spectrum in the complement of the pure point part. The known scaling relations are summarised, and some new findings are explained. 
View PDF on arXiv
9 Citations
Background Citations
3
Methods Citations
2

Figures and Tables from this paper

9 Citations

Scaling properties of the thue–morse measure

The classic Thue–Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many…

Bragg spectrum, K-theory and Gap Labelling of aperiodic solids

. The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute…

Bragg spectrum and Gap Labelling of aperiodic solids

The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute…

Recent progress in mathematical diffraction

A brief summary of recent developments in mathematical diffraction theory is given. Particular emphasis is placed on systems with aperiodic order and continuous spectral components. We restrict…

Lyapunov exponents for binary substitutions of constant length

A method of confirming the absence of absolutely continuous diffraction via the positivity of Lyapunov exponents derived from the corresponding Fourier matrices is presented, which provides an…

Aperiodic crystals and beyond.

  • U. Grimm
  • Materials Science
    Acta crystallographica Section B, Structural science, crystal engineering and materials
  • 2015
Considering a number of recently investigated toy systems, with particular emphasis on non-crystalline ordered structures, the limits of the current definition of crystals are explored.

A natural probability measure derived from Stern's diatomic sequence

Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We…

Statistical Approach to Diffraction of Periodic and Non-Periodic Crystals—Review

In this paper, we show the fundamentals of statistical method of structure analysis. Basic concept of a method is the average unit cell, which is a probability distribution of atomic positions with…

Scaling Properties of the Thue–Morse Measure: A Summary

References

SHOWING 1-10 OF 36 REFERENCES

The singular continuous diffraction measure of the Thue–Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue–Morse chain, in its realization as a binary sequence with values in {±1}. We…

On the generalized dimensions for the Fourier spectrum of the Thue-Morse sequence

We present explicit relations for the generalized dimensions of the spectral measure of the Thue-Morse symbolic sequence for positive integer values of q. Each is expressed through the eigenvalue of…

On the correlation dimension of the spectral measure for the thue-morse sequence

Using the relationship between the decay rate of autocorrelation and the characteristics of singular Fourier spectra, we show that the correlation dimension of the spectral measure for the infinite…

Spectral and topological properties of a family of generalised Thue-Morse sequences

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is know as the Devil's staircase. The support of the Cantor measure is a set of zero…

How should one define a (weak) crystal?

We compare two proposals for the study of positional long-range order: one in terms of the spectrum of the translation operator, the other in terms of the Fourier spectrum. We point out that only the…

On the dimensions of the spectral measure of symmetric binary substitutions

Substitution rules defined on the binary alphabet and invariant under the interchange of both participating symbols are considered and exact relations are derived which express the correlation dimension of the multifractal spectral measure in terms of the entries in the substitution pattern.

Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures

The authors propose to use multifractal analysis in reciprocal space as a tool to characterise, in a statistical and global sense, the nature of the Fourier transform of geometrical models for atomic…

Multidimensional constant-length substitution sequences

Squirals and beyond: substitution tilings with singular continuous spectrum

Abstract The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of ${ \mathbb{Z} }^{2} $. In particular,…

Generalized Morse sequences

SummaryA method for construction of almost periodic points in the shift space on two symbols is developed, and a necessary and sufficient condition is given for the orbit closure of such a point to…