Intrinsic metric on graded graphs,standardness,and invariant measures

Abstract

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.