Universal adic approximation, invariant measures and scaled entropy

We define an infinite graded graph of ordered pairs and a canonical action of the group $\mathbb{Z}$ (the adic action) and of the infinite sum of groups of order two $\mathcal{D}=\sum_1^{\infty} \mathbb{Z}/2\mathbb{Z}$ 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.