We consider topological dynamical systems over $${\mathbb {Z}}$$ Z and, more generally, locally compact, $$\sigma $$ σ -compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of diffraction theory to associate an autocorrelation and a diffraction measure to any $$L^2$$ L 2 -function over such a dynamical system. This diffraction measure is shown to be the spectral measure of the function. If the group has a countable basis of the topology one can also exhibit the underlying autocorrelation by sampling along the orbits. Building on these considerations we then show how the spectral theory of dynamical systems can be reformulated via diffraction theory of function dynamical systems. In particular, we show that the diffraction measures of suitable factors provide a complete spectral invariant.

中文翻译：

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动力系统的谱理论作为采样函数的衍射理论

我们考虑了 $${\mathbb {Z}}$$ Z 上的拓扑动力系统，更一般地说，是局部紧致的 $$\sigma $$ σ -紧致阿贝尔群。我们将光谱理论和衍射理论联系起来。我们首先使用最近开发的衍射理论的一般框架将自相关和衍射测量与此类动态系统上的任何 $$L^2$$L 2 函数相关联。该衍射测量显示为函数的光谱测量。如果该组具有拓扑的可数基，则还可以通过沿轨道采样来展示潜在的自相关。基于这些考虑，我们然后展示了如何通过函数动力系统的衍射理论重新表述动力系统的谱理论。特别是，