Universal adic approximation, invariant measures and scaled entropy

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.Comment: 32 pp. Ref 3

Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph

“…Namely, we prove that all (aperiodic) automorphisms of a Lebesgue space can be realized on a single special graph endowed with an adic structure; it suffices to vary only a central measure µ on its path space. Earlier, a similar result was obtained for the class of automorphisms having the dyadic odometer as a quotient: all such automorphisms can be realized on the so-called graph of ordered pairs (for details, see [18]).…”

“…Arguing in this way, we arrive at the following "Borel" question, which was raised in [18]. Let X be a standard Borel space and T be an (aperiodic) Borel automorphism of X.…”

Entropy invariants of generic actions

“…Among the wide variety of the entropy notions (see, for example, [4], [11], [22], [24]) we consider slightly modified invariants of Kushnirenko [13].…”

On a Universal Borel Adic Space

“…The term "uniadic" derives from the words "universal" and "semi-dyadic," where the latter means that every vertex of level n, for n ≥ 1, has one or two edges coming to it from vertices of level n − 1. The predecessors of this graph are dyadic graphs: the graph of unordered pairs (see [8]) and the graph of ordered pairs (see [9]); each of them is of considerable interest.…”