Universal adic approximation, invariant measures and scaled entropy

TL;DR

This paper introduces a universal adic action on an infinite graded graph's path space, demonstrating its isomorphism with ergodic actions of certain groups with invariant measures, and explores related problems.

Contribution

It constructs a universal adic action on an infinite graph and proves its isomorphism with a broad class of ergodic group actions, advancing the understanding of invariant measures and entropy.

Findings

Actions are universal for groups nd .

Every ergodic action with invariant measure is isomorphic to the adic action.

The paper explores related open problems.

Abstract

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.