Universal adic approximation, invariant measures and scaled entropy

摘要

We define an infinite graded graph of ordered pairs and a canonical action of the group Z (the adic action) and of the infinite sum of groups of order two D = Sigma(infinity)(1) Z/2Z on the path space of the graph. It is proved that these actions are universal for both groups in the following sense: every ergodic action of these groups with invariant measure and binomial generator, multiplied by a special action (the 'odometer'), is metrically isomorphic to the canonical adic action on the path space of the graph with a central measure. We consider a series of related problems.