Conjugacies of model sets

Abstract

Let $M$ be a model set meeting two simple conditions: (1) the internal space $H$ is a product of $R^n$ and a finite group, and (2) the window $W$ is a finite union of disjoint polyhedra. Then any point pattern with finite local complexity (FLC) that is topologically conjugate to $M$ is mutually locally derivable (MLD) to a model set $M'$ that has the same internal group and window as $M$, but has a different projection from $H \times R^d$ to $R^d$. In cohomological terms, this means that the group $H^1_{an}(M,R)$ of asymptotically negligible classes has dimension $n$. We also exhibit a counterexample when the second hypothesis is removed, constructing two topologically conjugate FLC Delone sets, one a model set and the other not even a Meyer set.