Conjugacies of model sets

TL;DR

This paper studies the conjugacy relations of certain model sets with specific internal space and window conditions, showing how topological conjugacy implies mutual local derivability and exploring counterexamples when conditions are relaxed.

Contribution

It establishes conditions under which topologically conjugate FLC point patterns are MLD to model sets with the same internal group and window, and provides counterexamples when these conditions are not met.

Findings

Topologically conjugate FLC patterns are MLD to model sets with the same internal group and window.

The dimension of the asymptotically negligible classes group is equal to the dimension of the Euclidean part of the internal space.

Counterexamples show that relaxing conditions can lead to conjugate sets that are not model sets or even Meyer 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.