Maximal equicontinuous generic factors and weak model sets

TL;DR

This paper investigates the structure of regular model sets and introduces the concept of a maximal equicontinuous generic factor (MEGF) to better understand systems with irregular windows, providing new insights into their dynamical properties.

Contribution

The paper defines and characterizes the maximal equicontinuous generic factor for ergodic topological systems, extending the understanding of model sets with irregular windows.

Findings

The MEGF exists for ergodic topological systems and is characterized by the regional proximal relation.

The MEGF is trivial if and only if the system is topologically weakly mixing.

$(G\times H)/\mathcal{L}$ is the MEGF for orbit closures of weak model sets with aperiodic Haar regular windows.

Abstract

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice $\mathcal{L}\subset G\times H$ and compact and aperiodic window $W\subseteq H$, have the maximal equicontinuous factor (MEF) $(G\times H)/\mathcal{L}$, if the window is toplogically regular. This picture breaks down completely, when the window has empty interior, in which case the MEF is always trivial, although $(G\times H)/\mathcal{L}$ continues to be the Kronecker factor for the Mirsky measure. As this situation occurs for many interesting examples like the square-free numbers or the visible lattice points, there is some need for a slightly weaker concept of topological factors that is still strong enough to capture basic properties of the system. Here we propose to use the concept of a generic factor \cite{HuangYe2012} for this purpose. For so called ergodic topological dynamical systems we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. For such systems we also show that the MEGF is trivial if and only if the system is topologically weakly mixing. This part of the paper profits strongly from previous work by McMahon \cite{McMahon1978} and Auslander \cite{Auslander1988}. In Section 3 we show that $(G\times H)/\mathcal{L}$ is indeed the MEGF of the orbit closure of each weak model set with an aperiodic Haar regulaar window, and in Section 4 we apply this fact to give an alternative proof of the finding \cite{Mentzen2017} that the centralizer of any $\mathcal{B}$-free dynamical system of Erd\"os type is trivial.