分次图的内度量、标准性和不变测度
Intrinsic metric on graded graphs,standardness,and invariant measures
摘要：
我们定义了一个一般的概念，光滑不变（中心）遍历测度的路径空间的$N$ -分次图（Bratteli图）。它是基于路径空间中的尾部过滤的标准性的概念，并且平滑性准则使用可以在这些图的顶点集合上规范地定义的所谓的内在度量。在作者已知的许多情况下，如Pascal图，Young图，配置空间，所有遍历中心测度都是光滑的（在这种情况下，我们说图是光滑的）。但即使在这些情况下，内在度量也远非显而易见，与“自然”度量不一致。我们应用和推广的过滤理论的作者在过去的四十年中开发的情况下，尾部过滤，特别是，引入了标准过滤的概念，作为推广的情况下，半齐次过滤的概念的标准齐次（并矢）过滤在这个理论的意义。起着至关重要的作用的新概念的\textbf{intrinsic semi-metric on the set of vertices of a graph}和正规路径的概念，这使我们能够完善的遍历方法的光滑措施的情况下。在未来，我们将应用这种新的方法的理论，不变测度在组合，遍历论，和理论的C $^*$ -代数。
英文摘要：
We define a general notion of a smooth invariant (central) ergodic measure on the space of paths of an $N$-graded graph (Bratteli diagram). It is based on the notion of standardness of the tail filtration in the space of paths, and the smoothness criterion uses the so-called intrinsic metric which can be canonically defined on the set of vertices of these graphs. In many cases known to the author, like the Pascal graph, the Young graph, the space of configurations, all ergodic central measures are smooth (in this case, we say that the graph is smooth). But even in these cases, the intrinsic metric is far from being obvious and does not coincide with the "natural" metric. We apply and generalize the theory of filtrations developed by the author during the last forty years to the case of tail filtrations and, in particular, introduce the notion of a standard filtration as a generalization to the case of semi-homogeneous filtrations of the notion of a standard homogeneous (dyadic) filtration in the sense of that theory. The crucial role is played by the new notion of \textbf{intrinsic semi-metric on the set of vertices of a graph} and the notion of regular paths, which allows us to refine the ergodic method for the case of smooth measures. In future, we will apply this new approach to the theory of invariant measures in combinatorics, ergodic theory, and the theory of C$^*$-algebras.
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