Self-Similarities and Invariant Densities for Model Sets

1998/09/03 by Michael Baake, Robert V. Moody
Physics and Astronomy · Mathematics · #math-ph #cond-mat #math.MP

paper · pdf

published as Algebraic Methods in Physics (eds. Y. Saint-Aubin and L. Vinet), Springer, New York (2001), pp. 1-15 · 15 pages, 2 figures, to appear in: Algebraic Methods and Theoretical Physics (ed. Y. St. Aubin)

arxiv created 1998/09/03 · arxiv updated 2009/11/30

Abstract

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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