Characterizations of model sets by dynamical systems

Michael Baake, Daniel Lenz, Robert V. Moody

Published 2005-11-27, updated 2007-02-15Version 2

It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $\beta$ and then relate the properties of $\beta$ to ones of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.