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

@article{Godrche1990MultifractalAI,
  title={Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures},
  author={Claude Godr{\`e}che and Jean-Marc Luck},
  journal={Journal of Physics A},
  year={1990},
  volume={23},
  pages={3769-3797}
}
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 structures. This approach is especially adequate for shedding some new light on a class of structures introduced recently, which exhibit 'singular scattering'. Using the language of measure theory, the Fourier intensity of these models is presumably singular continuous, and therefore represents an… 
The nature of the atomic surfaces of quasiperiodic self-similar structures
Quasiperiodic self-similar chains generated by substitutions (i.e. deterministic concatenation rules) and their diffraction spectra are analysed in a systematic fashion, from the viewpoint of the
Renormalisation of Pair Correlations and Their Fourier Transforms for Primitive Block Substitutions
For point sets and tilings that can be constructed with the projection method, one has a good understanding of the correlation structure, and also of the corresponding spectra, both in the dynamical
A study of the structure factor of Thue - Morse and period-doubling chains by wavelet analysis
TLDR
This example shows that wavelet analysis could become an efficient tool for studying the Fourier transform of aperiodic systems by computing the scaling exponents and finding the wavevector associated with them.
Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings
Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot-Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point
Applications of group cohomology to the classification of quasicrystal symmetries
In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. After the discovery of quasicrystals in 1984, Mermin and
Real Space Structure Factor for Different Quasicrystals
The statistical approach to the description of aperiodic structures using the concept of the so called Average Unit Cell (AUC) is presented. The use of this method is shown for 1D, 2D, and 3D
On superimposed dynamical multifractals
Dynamically generated multifractal measures are a generic model for the observation of fractal structures in nature. We investigate the effects obtained from the superposition of such structures. In
...
...

References

SHOWING 1-10 OF 12 REFERENCES
Scaling properties of a structure intermediate between quasiperiodic and random
We consider a one-dimensional structure obtained by stringing two types of “beads” (short and long bonds) on a line according to a quasiperiodic rule. This model exhibits a new kind of order,
Quasiperiodicity and types of order; a study in one dimension
In order to characterise the interplay between quasiperiodicity and order in one dimension, the authors consider sequences of 0 and 1 generated by a circle map. These sequences, which generalise the
A structure intermediate between quasi-periodic and random
We consider an infinite chain of atoms, where the bond lengths between neighbouring sites take two values, according to a quasi-periodic rule, associated to a circle map with an irrational rotation
Number Theory and Physics: Proceedings of the Winter School, Les Houches, France, March 7-16, 1989
I Conformally Invariant Field Theories, Integrability, Quantum Groups.- Z/NZ Conformal Field Theories.- Affine Characters and Modular Transformations.- Conformal Field Theory on a Riemann Surface.-
Incommensurate structure with no average lattice : an example of a one-dimensional quasicrystal
We study the ground state of a simple one-dimensional model describing an incommensurate modulation of the vacancy density of a periodic lattice. We show that this structure, through its Fourier
Structure and electronic properties of Thue-Morse lattices.
We study a one-dimensional system which is neither periodic, quasiperiodic, nor random. We find that the structure factor of this system consists of a set of peaks whose heights scale with L, the
Substitution dynamical systems, spectral analysis
The Banach Algebra (T).- Spectral Theory of Unitary Operators.- Spectral Theory of Dynamical Systems.- Dynamical Systems Associated with Sequences.- Dynamical Systems Arising from Substitutions.-
Summation Formulae for Substitutions on a Finite Alphabet
Let A be a finite alphabet, σ a substitution over A, (un) n∈N a fixed point for σ, and for each a∈ A, f(a) a real number; $$ {s^f}\left( n \right)\sum\limits_{n\, \leqslant N} {F\left( {{u_i}}
...
...