The theory of filtrations of subalgebras, standardness, and independence

Abstract

In memory of my friend Kolya K. (1933Kolya K. ( -2014 "Independence is the best quality, the best word in all languages."J. Brodsky. From a personal letter. AbstractThe survey is devoted to the combinatorial and metric theory of filtrations, i. e., decreasing sequences of σ-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of standardness, plays the role of a generalization of the notion of the independence of a sequence of random variables. We discuss the possibility of obtaining a classification of filtrations, their invariants, and various links to problems in algebra, dynamics, and combinatorics.Bibliography: 101 titles.

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.…”

“…The proof is based on the construction of a socalled basic Borel filtration of a given automorphism. For more details on the history of the problem, see [8,10].…”

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

“…This paper deals with applications of the theory of metric filtrations (see [16] and the references therein) to uniform approximation of automorphisms of measure spaces and the analysis of adic transformations in path spaces of graphs. Conceptually, it is closely related to the first author's work on dyadic and homogeneous sequences of measurable partitions (= filtrations), standardness, the "scale" metric invariant, etc.…”

“…Combinatorial invariants are invariants of all finite fragments of filtrations, i.e., invariants of periodic approximations; they are described below and represent some measures on the space of hierarchies on the group Z. The prospect of obtaining an efficient combinatorial classification of filtrations described below was observed in [16], but the fact that this classification problem has indeed turned out to be tame gives hope for further classifications.…”

“…Here we suggest constructions of universal graphs on which every automorphism can be realized as an adic shift. The study of specific automorphisms has already begun (see the survey [1], and also [2]), but, according to the conclusion of the paper [16] (see also below), the problem of classification of combinatorially definite filtrations is tame, i.e., there is a manageable space of classes, or orbits, or complete metric invariants of such filtrations (see the definition of combinatorial schemes of hierarchies below). This gives a chance to obtain a reasonable combinatorial classification of measure-preserving automorphisms.…”