# Self-Similarities and Invariant Densities for Model Sets

@article{Baake1998SelfSimilaritiesAI,
title={Self-Similarities and Invariant Densities for Model Sets},
author={Michael Baake and Robert V. Moody},
journal={arXiv: Mathematical Physics},
year={1998},
pages={1-15}
}
• Published 3 September 1998
• Mathematics
• arXiv: Mathematical Physics
Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prove that invariant densities exist and that they produce absolutely continuous invariant measures in internal space. We study the invariant densities and their relationships to diffraction, continuous…
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## References

SHOWING 1-10 OF 22 REFERENCES

• Mathematics
• 1998
Model sets (also called cut and project sets) are generalizations of lattices, and multi-component model sets are generalizations of lattices with colourings. In this paper, we study
• Computer Science
• 1998
It is shown that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background.
• Mathematics
• 1997
We present a mathematical construction of n-dimensional quasicrystals by starting from its symmetry group, an arbitrary finite group, and from its local structure described by using a finite union of
Preface. Knotted Tilings C.C. Adams. Solution of the Coincidence Problem in Dimensions d smaller than or equal to 4 M. Baake. Self-Similar Tilings and Patterns Described by Mappings C. Bandt. Delone
• Mathematics, Chemistry