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, when the window has empty interior, in which case the MEF is trivial, although $ (G\times H)/ {\mathcal L} $ 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 ($ E $-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 $ {\mathcal B} $-free dynamical system of Erdős type is trivial.

中文翻译：

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最大等连续泛型因子和弱模型集

由协紧凑晶格$ {\ mathcal L} \ subset G \ times H $和紧实和非周期性窗口$ W \ subseteq H $给定的割投影方案生成的常规模型集的轨道闭合具有如果窗口在拓扑上规则，则最大等连续因子（MEF）$（G \ H）/ {\ L L} $。当窗口的内部为空时，此图片会分解，在这种情况下，MEF是微不足道的，尽管$（G \ times H）/ {\ mathcal L} $仍然是Mirsky度量的Kronecker因子。由于发生在许多有趣的例子中，例如无平方数或可见晶格点，因此需要一个较弱的拓扑因子概念，例如通用因子[24]。对于具有有限不变量度且具有全力支持的拓扑动力学系统（$ E $ -systems），我们证明了一个最大等连续通用因子（MEGF）的存在，并根据区域近端关系对其进行了表征。本文的这一部分从McMahon [33]和Auslander [2]。在第3和第4节中，我们确定了弱模型集的轨道闭合的MEGF，并将该结果用于事实的替代证明（泛化）[34]任何Erdős类型的无$ {\ mathcal B} $动力系统的扶正器都是微不足道的。