# 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.

## 9 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES

### The singular continuous diffraction measure of the Thue–Morse chain

- Physics
- 2008

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

- Mathematics
- 1999

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

- Mathematics
- 1997

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

- Mathematics
- 2012

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?

- Mathematics
- 1992

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

- Computer Science
- 2002

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

- Physics
- 1990

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…

### Squirals and beyond: substitution tilings with singular continuous spectrum

- MathematicsErgodic Theory and Dynamical Systems
- 2013

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

- Mathematics
- 1968

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…