Maximal equicontinuous generic factors and weak model sets

Abstract

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice \begin{document}$ {\mathcal L}\subset G\times H $\end{document} and compact and aperiodic window \begin{document}$ W\subseteq H $\end{document} , have the maximal equicontinuous factor (MEF) \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} , if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [ 24 ]. For topological dynamical systems that possess a finite invariant measure with full support ( \begin{document}$ E $\end{document} -systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [ 33 ] and Auslander [ 2 ]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [ 34 ] that the centralizer of any \begin{document}$ {\mathcal B} $\end{document} -free dynamical system of Erdős type is trivial.