Self-Similarities and Invariant Densities for Model Sets

Details Self-Similarities and Invariant Densities for Model Sets Self-Similarities and Invariant Densities for Model Sets

1998-09-03

arxiv.org/abs/math-ph/9809006v1

Algebraic Methods in Physics (eds. Y. Saint-Aubin and L. Vinet), Springer, New York (2001), pp. 1-15

Physics

Mathematical Physics

15 pages, 2 figures, to appear in: Algebraic Methods and Theoretical Physics (ed. Y. St. Aubin)

Scientific paper

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 refinement operators, and Hutchinson measures.

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